New coincidence and common fixed point theorems

@Volume 10 • Number 1 • 2009

Editors-in-Chief:

Salvador Romaguera - Manuel Sanchis

Published by

Instituto de Matematica Pura y Aplicada

ISSN 1989-4147

Editorial Board

Topological Dynamics: Francisco Balibrea, Departamento de Matematicas,Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain.e-mail: [email protected]

Function Algebras and Function Spaces: Jose L. Blasco, Departamento deAnalisis Matematico, Facultad de Matematicas, Universidad de Valencia E.G.,Doctor Moliner 50, 46100 Burjasot, Spain.e-mail: [email protected]

Mathematical Economics: Juan Candeal, Departamento de Analisis Economi-co, Universidad de Zaragoza, Gran Via 2-4, 50005 Zaragoza, Spain.e-mail: [email protected]

Fixed Point Theory, Fuzzy Topology: Valentın Gregori, Escuela Universitariade Gandia, Universidad Politecnica de Valencia, 46730 Grau de Gandia, Va-lencia, Spain.e-mail: [email protected]

Ordered Topological Structures: Esteban Indurain, Departamento de Matematicae Informatica, Universidad Publica de Navarra, Pamplona, Spain.e-mail: [email protected]

Hyperspaces, (Quasi-)Uniformities, Bitopologies: Salvador Romaguera, Es-cuela de Caminos, Departamento de Matematica Aplicada, Universidad Politecnicade Valencia, 46071 Valencia, Spain.e-mail: [email protected]

Topological Methods in Functional Analysis: Enrique A. Sanchez-Perez, Es-cuela de Caminos, Departamento de Matematica Aplicada, Universidad Poli-tecnica de Valencia, 46071 Valencia, Spain.e-mail: [email protected]

Topological Algebra, Extensions of Spaces: Manuel Sanchis, Departamentode Matematicas, Campus del Riu Sec, s/n, ESTCE, Universitat Jaume I, 12071Castellon, Spain.e-mail: [email protected]

Topology and Computer Science: Michel Schellekens, Department of Com-puter Science, National University of Ireland, Cork, Ireland.e-mail: [email protected]

iii

Technical Editors

J. J. Font, L. Garcıa-Raffi, J. Rodrıguez-LopezDepartamento de Matematica Aplicada

Universidad Politecnica de ValenciaCamino de Vera s/n

46022 ValenciaSpain

http://agt.webs.upv.es

[email protected]

iv

Contents

Almost cl-supercontinuous functions. By J. K. Kohli and D.

Singh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Embedding into discretely absolutely star-Lindelof spaces

II. By Y.-K. Song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Best proximity pair theorems for relatively nonexpansive

mappings. By V. Sankar Raj and P. Veeramani . . . . . . . . . . . . . . 21

∗-half completeness in quasi-uniform spaces. By A. Andrikopou-

los . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Condensations of Cp(X) onto σ-compact spaces. By V. V. Tkachuk 39

Pointwise convergence and Ascoli theorems for nearness

spaces. By Z. Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

F-supercontinuous functions. By J. K. Kohli, D. Singh and J.

Aggarwal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Arnautov’s problems on semitopological isomorphisms. By

D. Dikranjan and A. Giordano Bruno . . . . . . . . . . . . . . . . . . . . . . . . 85

New coincidence and common fixed point theorems. By S.

L. Singh, A. Hematulin and R. Pant . . . . . . . . . . . . . . . . . . . . . . . . . 121

Well-posedness, bornologies, and the structure of metric

spaces. By G. Beer and M. Segura . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Topologies on function spaces and hyperspaces. By D. N.

Georgiou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

v

@ Applied General Topology

c© Universidad Politecnica de Valencia

Volume 10, No. 1, 2009

pp. 1-12

Almost cl-supercontinuous functions

J. K. Kohli and D. Singh

Abstract. Reilly and Vamanamurthy introduced the class of‘clopen maps’ (≡ ‘cl-supercontinuous functions’). Subsequently gen-eralizing clopen maps, Ekici defined and studied almost clopen maps(≡ almost cl-supercontinuous functions). Continuing in the spirit ofEkici, here basic properties of almost clopen maps are studied. Behav-ior of separation axioms under almost clopen maps is elaborated. Theinterrelations between direct and inverse transfer of topological prop-erties under almost clopen maps are investigated. The results obtainedin the process generalize, improve and strengthen several known resultsin literature including those of Ekici, Singh, and others.

2000 AMS Classification: Primary: 54C05, 54C10; Secondary: 54D10,54D15, and 54D 20.

Keywords: almost clopen map, almost cl-supercontinuous function, (almost)z-supercontinuous function, clopen almost closed graphs, almost zero dimen-sional space, hyperconnected space.

1. Introduction

Variants of continuity occur in almost all branches of mathematics and ap-plications of mathematics. The strong variants of continuity with which weshall be dealing in this paper include strongly continuous functions introducedby Levine [13], perfectly continuous functions considered by Noiri ([18], [19]),clopen maps (≡ cl-supercontinuous functions) defined by Reilly and Vamana-murthy [21], and studied by Singh [26], z-supercontinuous functions initiatedby Kohli and Kumar [12], and supercontinuous functions introduced by Munshiand Bassan [16]. The variants of continuity which are independent of continu-ity and will be dealt with in this paper include regular set connected functions(≡ almost perfectly continuous functions) defined by Dontchev, Ganster andReilly [3], almost clopen maps (≡ almost cl-supercontinuous functions) studiedby Ekici [4], almost z-supercontinuous functions [11] and δ-continuous func-tions defined by Noiri [17]. Moreover, the weak forms of continuity which will

2 J. K. Kohli and D. Singh

crop up in our discussion include almost continuous functions due to Singal andSingal [24], θ-continuous functions [5], quasi θ-continuous functions [20], weaklycontinuous functions [14], faintly continuous functions [15], Dδ-continuous func-tions [9], z-continuous functions [23], and others.

The purpose of this paper is to study properties of almost cl-supercontinuousfunctions (≡ almost clopen maps). In the process we generalize, improve andrefine several known results in the literature including those of Ekici [4], Singh[26], and others.

Section 2 is devoted to basic definitions, preliminaries and nomenclature. InSection 3 of this paper we study basic properties of almost cl-supercontinuousfunctions. It is shown that (i) almost cl-supercontinuity is preserved under theexpansion of range as well as under the shrinking of range if f(X) is δ-embeddedin Y ; (ii) A mapping into a product space is almost cl-supercontinuous if andonly if its composition with each projection map onto the co-ordinate spaceis almost cl-supercontinuous; (iii) If X is almost zero-dimensional, then fis almost cl-supercontinuous if and only if the graph function is almost cl-supercontinuous.

Section 4 is devoted to the behavior of separation axioms under almost cl-supercontinuous functions wherein interrelations between direct and inversetransfer of separation properties are investigated. In the process we generalizeand considerably improve upon certain results of Ekici [4], and Singh [26].

In Section 5, we interrelate (almost) cl-supercontinuity and connectedness.In the process we prove the existence and nonexistence of certain (almost)cl-supercontinuous functions. In Section 6, we consider clopen almost closedgraphs and obtain refinements of certain results of Ekici [4].

2. Preliminaries and basic definitions

2.1. Nomenclature. Reilly and Vamanamurthy [21] call a function clopencontinuous if for each open set V containing f(x) there is a clopen (closedand open) set U containing x such that f(U) ⊂ V . Similarly, Ekici [4] calls afunction almost clopen if for each x ∈ X and each regular open set V containingf(x) there is a clopen set U containing x such that f(U) ⊂ V . Moreover,Dontchev, Ganster and Reilly [3] call a function regular set connected if f−1(V )is clopen in X for every regular open set V in Y .

However, as was also pointed out in [26] that in the topological folklore thephrase “clopen map” is used for the functions which map clopen sets to opensets and hence therein the “clopen continuous maps” of Reilly and Vamana-murthy are renamed as “cl-supercontinuous functions”, a better nomenclaturesince it represents a strong form of supercontinuity introduced by Munshi andBassan [16]. In the same sprit in this paper we rename “almost clopen maps”studied by Ekici [4] as “almost cl-supercontinuous functions” and ”regular setconnected functions” defined by Dontchev, Ganster and Reilly [3] as “almostperfectly continuous functions”, respectively.

For the convenience of the reader and for the clarity of presentation we givehere the precise definitions of all these variants of continuity.

Almost cl-supercontinuous functions 3

Definition 2.2. A function f : X → Y from a topological space X into atopological space Y is said to be

(i) strongly continuous [13] if f(A) ⊂ f(A) for each subset A of X.(ii) perfectly continuous [18] if f−1(V ) is clopen in X for every open set

V ⊂ Y .(iii) almost perfectly continuous (≡ regular set connected [3]) if f−1(V ) is

clopen for every regular open set V in Y .(iv) cl-supercontinuous [26] (≡ clopen map [21]) if for each open set V con-

taining f(x) there is a clopen set U containing x such that f(U) ⊂ V .(v) almost cl-supercontinuous (≡ almost clopen map [4]) if for each x ∈

X and each regular open set V containing f(x) there is a clopen set Ucontaining x such that f(U) ⊂ V .

(vi) z-supercontinuous [12] if for each x ∈ X and each open set V containingf(x) there is a cozero set U containing x such that f(U) ⊂ V .

(vii) almost z-supercontinuous [11] if for each x ∈ X and each regular openset V containing f(x) there is a cozero set U containing x such thatf(U) ⊂ V .

(viii) supercontinuous [16] if for each x ∈ X and each open set V containingf(x) there is a regular open set U containing x such that f(U) ⊂ V .

(ix) δ-continuous [17] if for each x ∈ X and each regular open set V containingf(x) there is a regular open set U containing x such that f(U) ⊂ V .

(x) almost continuous [24] if for each x ∈ X and each regular open set Vcontaining f(x) there is an open set U containing x such that f(U) ⊂ V .

Remark 2.3. The original definitions of the concepts (v), (vii), (viii), (ix) and(x) in Definitions 2.2 are slightly different from the ones which first appearedin the literature but are equivalent to ones given here, and are the simplest andmost convenient to work with.

The following implications are immediate from the definitions and well known(or easily verified).

strongly continuous

⇓

perfectly continuous ⇒ almost perfectly continuous

⇓ ⇓

cl-supercontinuous ⇒ almost cl-supercontinuous

⇓ ⇓

z-supercontinuous ⇒ almost z-supercontinuous

⇓ ⇓

supercontinuous ⇒ δ-continuous

⇓ ⇓

continuous ⇒ almost continuous

However, it is well known that none of the above implications is reversible.


3. Basic properties of almost cl-supercontinuous functions

Definition 3.1. A set G in a topological space X is said to be cl-open [26]( δ-open [29]) if for each x ∈ G, there exist a clopen (regular open) set H suchthat x ∈ H ⊆ G, equivalently G is the union of clopen (regular open) sets. Thecomplement of a cl-open (δ-open) set is referred to as cl-closed ( δ-closed) set.

Theorem 3.2. Let f : X → Y and g : Y → Z be functions. Then the followingstatements are true.

(a) If f is cl-supercontinuous and g is continuous, then gf is cl-supercontinuous.(b) If f is cl-supercontinuous and g is almost continuous, then g f is almost

cl-supercontinuous.(c) If f is almost cl-supercontinuous and g is δ-continuous, then gf is almost

cl-supercontinuous.(d) If f is almost cl-supercontinuous and g is supercontinuous, then g f is

cl-supercontinuous.

Proof. The assertion (a) is due to Singh (see [26, Theorem 2.10]) and (b) is dueto Ekici [4, Theorem 13(2)].

To prove (c); let W ⊂ Z be a regular open set. Since g is δ-continuous,g−1(W ) is a δ-open set in Y , i.e. g−1(W ) =

⋃

α

Vα, where each Vα is a regular

open set in Y (see [17]). Since f is almost cl-supercontinuous, each f−1(Vα) iscl-open in X . Thus (g f)−1(W ) = f−1(g−1(W )) = f−1(

⋃

α

Vα) =⋃

α

f−1(Vα)

being the union of cl-open sets is cl-open in X and so g f is almost cl-supercontinuous.

To prove (d); let W be an open set in Z. Since g is supercontinuous, g−1(W )is δ-open set in Y , i.e. g−1(W )=

⋃

α

Vα, where each Vα is a regular open set in

Y (see[16]). Since f is almost cl-supercontinuous, f−1(Vα) is a cl-open set in Xfor each α. Thus (g f)−1(W )=f−1(g−1(W ))=f−1(

⋃

α

Vα)=⋃

α

f−1(Vα) being

the union of cl-open sets is cl-open in X . Hence g f is cl-supercontinuous.

Remark 3.3. The assertion (c) of Theorem 3.2 represents a simultaneousgeneralization of parts (1), (4), (5) and (6) of Theorem 13 of Ekici [4].

Theorem 3.4. Let Xα : α ∈ Λ be a cl-open cover of X. If for each αfα = f |Xα is almost cl-supercontinuous, then f is almost cl-supercontinuous.

Proof. Let V be a regular open subset of Y . Then f−1(V ) = ∪f−1α (V ) : α ∈

Λ. Since each fα is almost cl-supercontinuous, each f−1α (V ) is cl-open in Xα

and hence in X . Thus f−1(V ) being the union of cl-open sets is cl-open andso f is almost cl-supercontinuous.

Remark 3.5. Since every clopen set is cl-open, Theorem 3.4 is an improvementof Theorem 11 of Ekici [4].

Our next result gives a sufficient condition for the preservation of almost cl-supercontinuity under the shrinking of range. First we formulate the conceptof a δ-embedded set which seems to be of considerable significance in itself.


Definition 3.6. A subset S of a space X is said to be δ-embedded in X if everyregular open set in S is the intersection of a regular open set in X with S orequivalently every regular closed set in S is the intersection of a regular closedset in X with S.

Theorem 3.7. Let f : X → Y be an almost cl-supercontinuous function. Iff(X) is δ-embedded in Y , then f : X → f(X) is almost cl-supercontinuous.

Proof. Let V1 be a regular open set in f(X). Since f(X) is δ-embedded inY , there exists a regular open set V in Y such that V1 = V ∩ f(X). Again,since f is almost cl-supercontinuous, f−1(V ) is cl-open in X . Now f−1(V1) =f−1(V ∩ f(X)) = f−1(V ) ∩ f−1(f(X)) = f−1(V ) and so f : X → f(X) isalmost cl-supercontinuous.

Remark 3.8. In contrast to Theorem 3.7, it is easily verified that almost cl-supercontinuity is preserved under the expansion of range. The following lemmadue to Singal and Singal [24] will be used in the sequel.

Lemma 3.9 ([24]). Let Xα : α ∈ Λ be a family of spaces and let X =∏

Xα

be the product space. If x = (xα) ∈ X and V is a regular open set containing x,then there exists a basic regular open set

∏Vα such that x ∈

∏Vα ⊂ V , where

Vα is a regular open set in Xα for each α ∈ Λ and Vα = Xα for all exceptfinitely many α1, α2 . . . αn ∈ Λ.

Our next result shows that a mapping into a product space is almost cl-supercontinuous if and only if its composition with each projection map ontoa co-ordinate space is almost cl-supercontinuous.

Theorem 3.10. Let fα : X → Xα : α ∈ Λ be a family of functions andlet f : X →

∏αεΛ Xα be defined by f(x) = (fα(x)) for each x ∈ X. Then

f is almost cl-supercontinuous if and only if each fα : X → Xα is almostcl-supercontinuous.

Proof. Let f : X →∏

αεΛ Xα be almost cl-supercontinuous. Since projectionmaps are δ-continuous, then in view of Theorem 3.2 (c) the composition fα =pα f , where pα denotes the projection of

∏αεΛ Xα onto αth-coordinate space

Xα, is almost cl-supercontinuous for each α.Conversely, suppose that each fα : X → Xα is almost cl-supercontinuous.

To show that the function f is almost cl-supercontinuous, it is sufficient toshow that f−1(V ) is cl-open for each regular open set V in the product space∏

αεΛ Xα. In view of Lemma 3.9, it is clear that each regular open set Vin the product space

∏Xα is the union of basic regular open sets of the

form∏

Vα where each Vα is regular open in Xα and Vα = Xα for eachα except finitely many indices α1, α2 . . . αn. Thus each basic regular openset in

∏Xα is the finite intersection of sub-basic regular open sets of the

form Vβ ×∏

α6=β Xα, where Vβ is a regular open set in Xβ. Since arbitraryunions and finite intersections of cl-open sets is cl-open, it suffices to prove thatf−1(S) is cl-open for every subbasic regular open set S in the product space∏

αεΛ Xα. Let Vβ×∏

α6=β Xα be a subbasic regular open set in∏

αεΛ Xα. Then


f−1(Vβ ×∏

α6=β Xα) = f−1(pβ−1(Vβ)) = f−1

β (Vβ) is cl-open in X . Hence f isalmost cl-supercontinuous.

Definition 3.11 ([7]). A space X is said to be almost zero dimensional atx ∈ X if for every regular open set V containing x there exists a clopen setU containing x such that U ⊂ V . The space X is said to be almost zerodimensional if it is almost zero dimensional at each x ∈ X.

Theorem 3.12 ([7]). A space X is almost zero dimensional if and only if eachregular open set in X is cl-open.

Theorem 3.13. Let f : X → Y be a function and g : X → X × Y , defined byg(x) = (x, f(x)) for each x ∈ X, be the graph function. Then g is almost cl-supercontinuous if and only if f is almost cl-supercontinuous and X is almostzero dimensional.

Proof. Let g : X → X × Y be almost cl-supercontinuous. Then in view ofTheorem 3.2 (c) it is immediate that the composition f = py g is almostcl-supercontinuous, where py is the projection from X × Y onto Y (see also[4,Theorem12]). To prove that X is almost zero dimensional, let U be a reguaropen set in X and let x ∈ U . Then U ×Y is a regular open set containing g(x).Since g is almost cl-supercontinuous, there exists a clopen set W containing xsuch that g(W ) ⊂ U × Y . Thus x ∈ W ⊂ U , which shows that U is a cl-openand so the space X is almost zero dimensional.

To prove sufficiency, let x ∈ X and let W be a regular open set containingg(x). By Lemma 3.9 there exist regular open sets U ⊂ X and V ⊂ Y suchthat (x, f(x)) ∈ U × V ⊂ W . Since X is almost zero dimensional, there existsa clopen set G1 in X containing x such that x ∈ G1 ⊂ U . Since f is almostcl-supercontinuous, there exists a clopen set G2 in X containing x such thatf(G2) ⊂ V . Let G = G1 ∩ G2. Then G is a clopen set containing x andg(G) ⊂ U × V ⊂ W . This proves that g is almost cl-supercontinuous.

4. Separation axioms

Definitions 4.1. A space X is said to be

(i) ultra Hausdorff [27] if for each pair of distinct points x and y in X thereexist disjoint clopen sets U and V containing x and y, respectively.

(ii) ultra T1 (≡clopen T1 [4]) if for each pair of distinct points x and y in Xthere exist clopen sets U and V containing x and y, respectively such thaty /∈ U and x /∈ V .

(iii) ultra T0-space if for each pair of distinct points x and y in X there existsa clopen set U containing one of the points x and y but not the other.

Proposition 4.2. For a topological space X the following statements are equiv-alent.

(a) X is an ultra Hausdorff space.(b) X is an ultra T1-space.(c) X is an ultra T0-space.


Proof. Clearly (a)⇒(b)⇒(c). To prove (c)⇒(a), let X be an ultra T0-spaceand let x, y be any two distinct points in X . Then there exists a clopen set Ucontaining one of the points x and y but not the other. To be precise assumethat x ∈ U Then U and X \ U are disjoint clopen sets containing x and y,respectively and so X is an ultra Hausdroff space.

Definitions 4.3. A topological space X is said to be

(i) δT1-space (≡ r-T1 space [4]) if for each pair of distinct points x and y inX there exist regular open sets U and V containing x and y, respectivelysuch that y /∈ U and x /∈ V .

(ii) δT0-space if for each pair of distinct points x and y in X there exists aregular open set containing one of the points x and y but not the other.

Hausdorff space ⇒ δT1-space ⇒ δT0-space

⇓ ⇓

T1-space ⇒ T0-space

Example 4.4. The real line with co-finite topology is a T1-space which is notδT0 and so not a δT1-space.

It is shown in [26] that if f : X → Y is a cl-supercontinuous injection intoa T0-space Y , then X is an ultra-Hausdorff space. In contrast, for an almostcl-supercontinuous injection we have the following.

Theorem 4.5. Let f : X → Y be an almost cl-supercontinuous injection. IfY is a δT0-space, then X is an ultra-Hausdorff space.

Proof. Let x1 and x2 be two distinct points in X . Then f(x1) 6= f(x2). SinceY is a δT0-space, there exists a regular open set V containing one of the pointsf(x1) or f(x2) but not the other. To be precise, assume that f(x1) ∈ V .Since f is an almost cl-supercontinuous function, there exists a clopen set Ucontaining x1 such that f(U) ⊂ V . Then U and X \U are disjoint clopen setscontaining x1 and x2 respectively and so X is ultra-Hausdorff.

Remark 4.6. The above theorem generalizes Theorems 20 and 22 of Ekici [4].

Further, Ekici ([4,Theorem23]) proved that the equalizer of two almost cl-supercontinuous functions into a Hausdorff space is closed. Here we obtain thefollowing stronger version.

Theorem 4.7. Let f, g : X → Y be almost cl-supercontinuous functions intoa Hausdorff space Y . Then the equalizer E = x ∈ X : f(x) = g(x) of thefunctions f and g is a cl-closed subset of X.

Proof. To prove that E is cl-closed, we shall show that X \ E is cl-open. Tothis end, let x ∈ X \ E. Then f(x) 6= g(x). Since Y is Hausdorff, there existdisjoint open sets U1 and V1 containing f(x) and g(x), respectively. ThenU = (U1)

0 and V = (V1)0 are disjoint regular open sets containing f(x) and

g(x), respectively. Since f and g are almost cl-supercontinuous functions, thereexist clopen sets G1 and G2 containing x such that f(G1) ⊂ U and g(G2) ⊂ V .


Then G = G1 ∩ G2 is a clopen set containing x. Since U and V are disjoint,clearly G ⊂ X \ E and so X \ E is cl-open.

The following theorem represents an strengthening of Theorem 24 of Ekici [4].

Theorem 4.8. Let f : X → Y be an almost cl-supercontinuous function intoa Hausdorff space Y . Then the set A = (x1, x2) ∈ X × X : f(x1) = f(x2) isa cl-closed subset of X × X.

Proof. Let (x, y) /∈ A. Then f(x) 6= f(y). Since Y is Hausdorff, there existdisjoint open sets U1 and V1 containing f(x) and f(y), respectively. ThenU = (U1)

0 and V = (V1)0 are disjoint regular open sets containing f(x) and

f(y), respectively. Since f is almost cl-supercontinuous, there exist clopensets G1 and G2 containing x and y, respectively such that f(G1) ⊂ U andf(G2) ⊂ V . Then G1 × G2 is a clopen subset of X × X containing (x, y) and(G1×G2)∩A = φ. Hence G1×G2 ⊂ (X×X)\A and so (X×X)\A is cl-openbeing the union of clopen sets. Thus A is a cl-closed subset of X × X .

Definitions 4.9. A space X is said to be

(i) almost regular [22] if for each regularly closed set F and each x /∈ F thereexist disjoint open sets U and V containing x and F , respectively.

(ii) mildly normal [25] if for every pair of disjoint regular closed sets A and Bthere exist disjoint open sets U and V containing A and B, respectively.

The following theorem shows that the hypothesis that “X is regular” inTheorem 27 of Ekici [4] is superfluous and hence can be omitted.

Theorem 4.10. Let f : X → Y be an almost cl-supercontinuous open bijection.Then Y is an almost regular space.

Proof. Let F be a regular closed subset of Y and let y be a point outsideF . Then f−1(y) ∩ f−1(F ) = φ and f−1(y) is a singleton. Since f is almostcl-supercontinuous, f−1(F ) is a cl-closed subset of X . Hence X \f−1(F ) is a cl-open subset of X containing f−1(y). So there exists a clopen set G containingf−1(y) such that G ⊂ X \ f−1(F ). Then G and X \G are disjoint clopen setscontaining f−1(y) and f−1(F ), respectively. Since f is an open bijection, f(G)and f(X \ G) are disjoint open sets containing y and F , respectively. So Y isan almost regular space.

Definitions 4.11. A space X is said to be weakly ∆-normal [2] (weakly θ-normal [8], [10]) if each pair of disjoint δ-closed (θ-closed) are contained indisjoint open sets.

The following theorem represents a significant improvement of Theorem 28of Ekici [4].

Theorem 4.12. Let f : X → Y be an almost cl-supercontinuous open bijectiondefined on a weakly θ-normal space X. Then Y is mildly normal.


Proof. Let A and B be disjoint regular closed subsets of Y . Since f is almostcl-supercontinuous, f−1(A) and f−1(B) are disjoint cl-closed subsets of X .Since every cl-closed set is θ-closed and since X is weakly θ-normal, there existdisjoint open sets U and V containing f−1(A) and f−1(B), respectively. Sincef is an open bijection, f(U) and f(V ) are disjoint open sets containing A andB, respectively and hence Y is mildly normal.

Corollary 4.13. Let f : X → Y be an almost cl-supercontinuous open bijectiondefined on a weakly ∆-normal space X. Then Y is mildly normal.

Corollary 4.14 (Ekici [4, Theorem 28]). If f is an almost cl-supercontinuousopen bijection from a normal space X onto a space Y , then Y is mildly normal.

Proof. Every normal space is a weakly θ-normal space.

5. Connectedness

Ekici [4] calls a space X almost connected if X can not be written as adisjoint union of two nonempty regular open sets.

We observe that a space is connected if and only if it can not be expressedas a disjoint union of two nonempty clopen sets and hence it can not be writtenas the disjoint union of two nonempty regular open sets. Thus the notion ofalmost connectedness introduced by Ekici is precisely connectedness.

Moreover, the hypothesis of Theorem 30 of Ekici [4] is too strong and canbe considerably weakened, since connectedness is preserved under functionssatisfying fairly mild continuity conditions. The known such weakest variant ofcontinuity is slight continuity [6]. A function f : X → Y is said to be slightlycontinuous if f−1(V ) is open in X for every clopen subset V of Y . Thusconnectedness is preserved under each of the following variants of continuitylisted in the following diagram, each of which is weaker than continuity exceptδ-continuity (which is independent of continuity).

almost cl-supercontinuous⇓

continuous δ-continuous [17]⇓ ⇓

almost continuous [24]⇓

θ-continuous [5]⇓ ⇓

quasi-θ-continuous [20] weakly continuous [14]⇓ ⇓

faintly continuous [15]⇓

Dδ-continuous [9]⇓

z-continuous [23]⇓

slightly-continuous [6]


Definition 5.1 ([27], [1]). A space X is said to be hyperconnected if everynonempty open set in X is dense in X.

Ekici [4] showed that an almost cl-supercontinuous image of a connectedspace is hyperconnected. In contrast, our next result shows that cl-supercontinuousimage of a connected space is indiscrete.

Theorem 5.2. Let f : X → Y be a cl-supercontinuous function from a con-nected space X onto a space Y . Then Y is an indiscrete space.

Proof. Suppose that Y is not indiscrete and let V 6= Y be an open set inY . Since f is cl-supercontinuous, by [26, Theorem 2.2] f−1(V ) is a nonemptyproper cl-open subset of X . So there exists a nonempty proper clopen subsetof X , contradicting the fact that X is connected.

Thus there exists no cl-supercontinuous function from a connected spaceonto a non indiscrete space. In contrast it is shown in [26, Theorem 4.9] thatthere exist no non constant cl-supercontinuous function from a connected spaceinto a T0-space.

6. Clopen almost closed graphs

Definition 6.1 ([4]). The graph G(f) of a function f : X → Y is said to beclopen almost closed if for each (x, y) /∈ G(f) there exists a clopen set U of xand a regular open set V containing y such that (U × V ) ∩ G(f) = φ.

The following theorem represents an improved version of Theorem 35 ofEkici [4] which was essentially proved by him. However, for the convenience ofthe reader we include its proof.

Theorem 6.2. Let f : X → Y be an injection such that its graph G(f) isclopen almost closed. Then X is ultra Hausdorff.

Proof. Let x, y ∈ X, x 6= y. Since f is an injection, (x, f(y)) /∈ G(f). In viewof almost closedness of the graph G(f), there exist a clopen set U of x anda regular open set V containing f(y) such that (U × V ) ∩ G(f) = φ. Thenf(U) ∩ V = φ and hence U ∩ f−1(V ) = φ. Therefore y /∈ U . Then U andX \ U are disjoint clopen sets containing x and y, respectively. Hence X isultra Hausdorff.

Finally, we point out that in the hypothesis of [4, Theorem 41, Part 3], it issufficient to assume X to be countably compact instead of compact.

The next result is an strengthening of Theorem 39 of Ekici [4] which wasessentially proved by him. However, for the sake of completeness and continuityof presentation, we include its proof.

Theorem 6.3. Let f : X → Y be a function such that the graph G(f) of fis clopen almost closed in X × Y . Then f−1(K) is cl-closed in X for everyN -closed subset K of Y .


Proof. Let K be an N -closed subset of Y . To prove that f−1(K) is cl-closed,we shall show that X \ f−1(K) is cl-open. To this end, let x ∈ X \ f−1(K).Then for each y ∈ K, (x, y) /∈ G(f). So there exists a clopen set Uy containingx and a regular open set Vy containing y such that (Uy × Vy) ∩ G(f) = φand hence f(Uy) ∩ Vy = φ. The collection Vy : y ∈ K is a cover of K byregular open sets in Y . So there exist finitely many y1, . . . , yn ∈ K such thatK ⊂

⋃n

i=1 Yyi. Let U =

⋂n

i=1 Uyi. Then U is a clopen set containing x such

that f(U) ∩ K = φ. Hence U ⊂ X \ f−1(K) and so U is cl-open being theunion of clopen sets.

References

[1] N. Ajmal and J. K. Kohli, Properties of hyperconnected spaces, their mapping into

Hausdorff space and embedding into hyperconnected spaces, Acta Math. Hungar. 60,no. 1-2 (1992), 41–49.

[2] A. K. Das, ∆-normal spaces and factorizations of normality, preprint.

[3] J. Dontchev, M. Ganster and I. Reilly, More on almost s-continuity, Indian J. Math. 41

(1999), 139–146.[4] E. Ekici, Generalizations of perfectly continuous, regular set connected and clopen func-

tions, Acta Math. Hungar. 107, no. 3 (2005), 193–205.[5] S. Fomin, Extensions of topological spaces, Annals of Math. 44 (1943), 471–480.[6] R. C. Jain, The role of regularly open sets in general topology, Ph.D. thesis, Meerut

Univ., Institute of Advanced Studies, Meerut, India (1980).[7] J. K. Kohli, Localization of topological propreties and certain generalizations of zero

dimensionality, preprint.[8] J. K. Kohli and A. K. Das, New normality axioms and decompositions of normality,

Glasnik Mat. 37, no. 57 (2003), 105–114.[9] J. K. Kohli and D. Singh, Between weak continuity and set connectedness, Studii Si

Cercetari Stintifice Seria Mathematica 15 (2005), 55–65.[10] J. K. Kohli and D. Singh, Weak normality properties and factorizations of normality,

Acta Math. Hungar. 110, no. 1–2 (2006), 67–80.[11] J. K. Kohli, D. Singh and R. Kumar, Generalizations of Z-supercontinuous functions

and Dδ-supercontinuous functions, Applied General Topology, to appear.[12] J. K. Kohli and R. Kumar, Z-supercontinuous functions, Indian J. Pure Appl. Math.

33, no. 7 (2002), 1097–1108.[13] N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67 (1960), 269.[14] N. Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly

68 (1961), 44–46.[15] P. E. Long and L. L. Herrington, Tθ-topology and faintly continuous functions, Kyung-

pook Math. J. 22 (1982), 7–14.[16] B. M. Munshi and D. S. Bassan, Super-continuous mappings, Indian J. Pure Appl. Math.

13 (1982), 229–236.[17] T. Noiri, On δ-continuous functions, J. Korean Math. Soc. 16 (1980), 161–166.[18] T. Noiri, Supercontinuity and some strong forms of continuity, Indian J. Pure. Appl.

Math. 15, no. 3 (1984), 241–250.[19] T. Noiri, Strong forms of continuity in topological spaces, Suppl. Rendiconti Circ. Mat.

Palermo, II 12 (1986), 107–113.[20] T. Noiri and V. Popa, Weak forms of faint continuity, Bull. Math. de la Soc. Math. de

la Roumanic 34, no. 82 (1990), 263–270.[21] I. L. Reilly and M. K. Vamanamurthy, On super-continuous mappings, Indian J.Pure.

Appl. Math. 14, no. 6 (1983), 767–772.[22] M. K. Singal and S. P. Arya, On almost regular spaces, Glasnik Mat. 4 (1969), 89–99.


[23] M. K. Singal and S. B. Nimse, Z-continuous mappings, The Mathematics Student 66,no. 1-4 (1997), 193–210.

[24] M. K. Singal and A. R. Singal, Almost continuous mappings, Yokohama Math. Jour. 16

(1968), 63–73.[25] M. K. Singal and A. R. Singal, Mildly normal spaces, Kyungpook Math. J. 3 (1973),

27–31.[26] D. Singh, cl-supercontinuous functions, Applied General Topology 8, no. 2 (2007), 293–

300.[27] R. Staum, The algebra of bounded continuous functions into nonarchimedean field, Pa-

cific J. Math. 50 (1974), 169–185.[28] L. A. Steen and J. A. Seeback, Jr., Counter Examples in Topology, Springer Verlag,

New York, 1978.[29] N. V. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl. 78, no. 2 (1968),

103–118.

Received September 2007

Accepted April 2008

J. K. Kohli (jk [email protected])Department of Mathematics, Hindu College, University of Delhi, Delhi 110 007,India

D. Singh ([email protected])Department of Mathematics, Sri Aurobindo College, University of Delhi-SouthCampus, Delhi 110 017, India



Volume 10, No. 1, 2009

pp. 13-20

Embedding into discretely absolutelystar-Lindelof spaces II

Yan-Kui Song∗

Abstract. A space X is discretely absolutely star-Lindelof if forevery open cover U of X and every dense subset D of X, there existsa countable subset F of D such that F is discrete closed in X andSt(F,U) = X, where St(F,U) =

⋃U ∈ U : U∩F 6= ∅. We show that

every Hausdorff star-Lindelof space can be represented in a Hausdorffdiscretely absolutely star-Lindelof space as a closed Gδ-subspace.

Keywords: star-Lindelof, absolutely star-Lindelof, centered-Lindelof

2000 AMS Classification: 54D20, 54G20

1. Introduction

By a space, we mean a topological space. A space X is absolutely star-

Lindelof (see [1]) (discretely absolutely star-Lindelof)(see [12, 13]) if for everyopen cover U of X and every dense subset D of X , there exists a countablesubset F of D such that St(F,U) = X (F is discrete and closed in X andSt(F,U) = X , respectively), where St(F,U) =

⋃U ∈ U : U ∩ F 6= ∅.

A space X is star-Lindelof (see [4, 7] under different names) (discretely star-

Lindelof)(see [9, 16]) if for every open cover U of X , there exists a count-able subset (a countable discrete closed subset, respectively) F of X such thatSt(F,U) = X. It is clear that every separable space and every discretely star-Lindelof space are star-Lindelof as well as every space of countable extent(inparticular, every countably compact space or every Lindelof space).

A family of subsets is centered (linked) provided every finite subfamily (everytwo elements, respectively) has nonempty intersection and a family is called

∗The author acknowledges support from the NSF of China Grant 10571081 and Projectsupported by the National Science Foundation of Jiangsu Higher Education Institutions ofChina (Grant No 07KJB110055)

14 Y.-K. Song

σ-centered (σ-linked) if it is the union of countably many centered subfami-lies(linked subfamilies, respectively). A space X is centered-Lindelof (linked-

Lindelof) (see [2, 3]) if for every open cover U of X has σ-centered (σ-linked)subcover.

From the above definitions, it is not difficult to see that every discretelyabsolutely star-Lindelof space is absolutely star-Lindelof, every discretely ab-solutely star-Lindelof space is discretely star-Lindelof, every absolutely star-Lindelof space is star-Lindelof, every star-Lindelof space is centered-Lindelof,every centered-Lindelof space is linked-Lindelof.

Bonanzinga and Matveev [2] proved that every Hausdorff (regular, Ty-chonoff) linked-Lindelof space can be represented as a closed subspace in aHausdorff (regular, Tychonoff, respectively)star-Lindelof space. They asked ifevery Hausdorff (regular, Tychonoff) linked-Lindelof space can be representedas a closed Gδ-subspace in a Hausdorff (regular, Tychonoff, respectively) star-Lindelof space. The author [10] gave a positive answer to their question. Theauthor [10] showed that every Hausdorff (regular, Tychonoff) linked-Lindelofspace can be represented as a closed Gδ-subspace in a Hausdorff (regular, Ty-chonoff, respectively) absolutely star-Lindelof space. The author [13] showedthat every separable Hausdorff (regular, Tychonoff, normal) star-Lindelof spacecan be represented in a Hausdorff (regular, Tychonoff, normal, respectively)discretely absolutely star-Lindelof space as a closed Gδ-subspace. The author[14] showed that every Hausdorff linked-Lindelof space can be represented ina Hausdorff discretely absolutely star-Lindelof space as a closed subspace andasked the following question:

Question 1.1. Is it true that every Hausdorff (regular, Tychonoff) linked-

Lindelof-space can be represented a closed Gδ-subspace in a Hausdorff (regular,

Tychonoff, respectively) discretely absolutely star-Lindelof space?

The purpose of this note is to give a construction showing every Hausdorfflinked-Lindelof space can be represented in a Hausdorff discretely absolutelystar-Lindelof space as a closed Gδ-subspace, which give a positive answer tothe above question in the class of Hausdorff spaces.

Throughout this paper, the cardinality of a set A is denoted by |A|. Letω denote the first infinite cardinal. For a cardinal κ, let κ+ be the smallestcardinal greater than κ. As usual, a cardinal is the initial ordinal and an ordinalis the set of smaller ordinals. When viewed as a space, every cardinal has theusual order topology. For each pair of ordinals α, β with α < β, we write[α, β] = γ : α ≤ γ ≤ β and (α, β) = γ : α < γ < β. Other terms andsymbols that we do not define will be used as in [5].

2. Embedding into discretely absolutely star-Lindelof spaces as

a closed Gδ-subspaces

First, we show that every Hausdorff star-Lindelof space can be representedin a Hausdorff discretely absolutely star-Lindelof space as a closed Gδ-subspace.

Embedding into discretely absolutely star-Lindelof spaces II 15

Recall the Alexandorff duplicate A(X) of a space X . The underlying set ofA(X) is X ×0, 1; each point of X ×1 is isolated and a basic neighborhoodof a point 〈x, 0〉 ∈ X×0 is of the from (U×0)∪((U×1)\〈x, 1〉), whereU is a neighborhood of x in X . It is well-known that A(X) is Hausdorff(regular,Tychonoff, normal) iff X is, A(X) is compact iff X is and A(X) is Lindelof iffX is.

Recall from [6] that a space X is absolutely countably compact (=acc) if forevery open cover U of X and every dense subset D of X , there exists a finitesubset F of D such that St(F,U) = X . It is not difficult to show that everyHausdorff acc space is countably compact (see [6]). In our construction, we usethe following lemma.

Lemma 2.1 ([8, 15]). If X is countably compact, then A(X) is acc. Moreover,

for any open cover U of A(X), there exists a finite subset F of X × 1 such

that A(X) \ St(F,U) ⊆ X × 0 is a finite subset consisting of isolated points

of X × 0.

Theorem 2.2. Every Hausdorff star-Lindelof space can be represented in a

Hausdorff discretely absolutely star-Lindelof space as a closed Gδ-subspace.

Proof. If |X | ≤ ω, then X is separable. The author [13] showed that everyseparable Hausdorff (regular, Tychonoff, normal) space can be represented inHausdorff (regular, Tychonoff, normal, respectively) discretely absolutely star-Lindelof space as a closed Gδ-subspace.

Let X be a star-Lindelof space with |X | > ω and let T be X with the discretetopology and let

Y = T ∪ ∞, where ∞ /∈ T

be the one-point Lindelofication of T . Pick a cardinal κ with κ ≥ |X |. Define

S(X, κ) = X ∪ (Y × κ+).

We topologize S(X, κ) as follows: Y × κ+ has the usual product topology andis an open subspace of S(X, κ), and a basic neighborhood of a point x of Xtakes the form

G(U, α) = U ∪ (U × (α, κ+)),

where U is a neighborhood of x in X and α < κ+. Then, it is easy to see thatX is a closed subset of S(X, κ) and S(X, κ) is Hausdorff if X is Hausdorff.

Let

R(X) = A(S(X, κ)) \ (X × 1).

Then, R(X) is Hausdorff if X is Hausdorff.Let

P(R(X)) = ((X × 0)× ω) ∪ (R(X) × ω)

be the subspace of the product of R(X) × (ω + 1). For each n ∈ ω, let

Xω = (X × 0)× ω and Xn = R(X) × n for each n ∈ ω.

Then,

P(R(X)) = Xω ∪ ∪n∈ωXn.

16 Y.-K. Song

From the construction of the topology of P(R(X)), it is not difficult to seethat X can be represented in P(R(X)) as a closed Gδ-subspace, since X ishomeomorphic to Xω, and P(R(X)) is Hausdorff if X is Hausdorff.

We show that P(R(X)) is discretely absolutely star-Lindelof. To this end,let U be an open cover of P(R(X)). Without loss of generality, we assumethat U consists of basic open sets of P(R(X)). Let S be the set of all isolatedpoints of κ+ and let

Dn1 = (((T × S) × 0)× n) ∪ (((T × κ+) × 1)× n),

Dn2 = ((∞ × κ+) × 1) × n and Dn = Dn1 ∪ Dn2 for each n ∈ ω.

If we put D = ∪n∈ωDn. Then, every element of D is isolated in P(R(X)),and every dense subset of P(R(X)) contains D. Thus, it is sufficient to showthat there exists a countable subset F of D such that

F is discrete closed in P(R(X)) and St(F,U) = P(R(X)).

For each x ∈ X , there exists a Ux ∈ U such that 〈〈x, 0〉, ω〉 ∈ Ux, Hencethere exist αx < κ+, nx ∈ ω and an open neighborhood Vx of x in X such that

((Vx × 0)× [nx, ω]) ∪ (A(Vx × (αx, κ+)) × [nx, ω)) ⊆ Ux.

If we put V = Vx : x ∈ X, then V is an open cover of X . For each n ∈ ω, letX ′

n = ∪x : nx = n, then X = ∪n∈ωX ′

n. For each x′ ∈ X \ X ′

n, there exists aUx′ ∈ U such that

〈〈x′, 0〉, n〉 ∈ Ux′ .

Hence, there exist αx′ < κ+ and an open neighborhood Vx′ of x′ in X suchthat

((Vx′ × 0) × n) ∪ (A(Vx′ × (αx′ , κ+)) × n) ⊆ Ux′ .

If we putVn = Vx : x ∈ X ′

n ∪ Vx′ : x′ ∈ X \ X ′

n.

Then, Vn is an open cover of X . Hence, there exists a countable subset F ′

n ofX such that X = St(F ′

n,U), since X is star-Lindelof. If we pick

αn0 > maxsupαx : x ∈ X ′

n, supαx′ : x′ ∈ X \ X ′

n.

Then, αn0 < κ+, since |X | ≤ κ.Let

Xn1 = ((X × 0)× n) ∪ (A(T × [αn0, κ+)) × n);

Xn2 = A(T × [0, αn0]) × n and Xn3 = A(∞ × κ+) × n).

Then,Xn = Xn1 ∪ Xn2 ∪ Xn3.

LetFn1 = ((F ′

n × αn0) × 1) × n.

Then, Fn1 is a countable subset of Dn1 and

((X ′

n × 0)× ω) ∪ Xn1 ⊆ St(Fn1,U),

since Ux ∩ Fn1 6= ∅ for each x ∈ X ′

n and Ux′ ∩ Fn1 6= ∅ for each x′ ∈ X \ X ′

n.Since Fn1 ⊆ Dn1 and Fn1 is countable. Then, Fn1 is closed in Xn by the


construction of the topology of Xn. Hence, Fn1 is closed in P(R(X)), since Xn

is open and closed in P(R(X)).On the other hand, since Y is Lindelof and [0, αn0] is compact, then Y ×

[0, αn0] is Lindelof, hence Xn2 = A(Y × [0, αn0]) × n is Lindelof. For eachα ≤ αn0, there exists a Uα ∈ U such that

〈〈〈∞, α〉, 0〉, n〉 ∈ Uα.

Hence, there exists an open neighborhood Vα of α in κ+ and an open neigh-borhood V ′

α of ∞ in Y such that

(A(V ′

α × Vα) × n) \ (〈〈〈∞, α〉, 1〉, n〉) ⊆ Uα.

Let V ′

n = Vα : α ≤ αn0. Then, V ′

n is an open cover of [0, αn0]. Hence,there exists a finite subcover Vα1

, Vα2, ...Vαm

, since [0, αn0] is compact. Let

Tn = ∪T \ V ′

αi: i ≤ m.

Then, Tn is a countable subset of T . For each i ≤ m, we pick xi ∈ Dn ∩ Uαi.

Let F ′

n2 = xi : i ≤ m. Then, F ′

n2 is a finite subset of Dn and

(((∞× [0, αn0])× 0)× n)∪ (A((T \ Tn)× [0, αn0])× n) ⊆ St(F ′

n2,U).

For each t ∈ Tn, since t× [0, αn0] is compact, then A(t× [0, αn0])×nis compact, hence there exists a finite subset Ft of Dn such that

A(t × [0, α0]) × n ⊆ St(Ft,U).

Let F ′′

n2 = ∪Ft : t ∈ Tn. Then, F ′′

n2 is countable, since Tn is countable. SinceF ′′

n2 ∩ (A(Y × α) × n) is countable for each α < κ+ and F ′′

n2 ∩ (A(t ×κ+)×n is finite for each t ∈ T , then F ′′

n2 is closed in Xn by the constructionof the topology of Xn, hence Fn2 is closed in P(R(X)), since Xn is open closedin P(R(X)). By the definition of F ′′

n2, we have

A(Tn × [0, αn0]) × n ⊆ St(F ′′

n2,U).

If we put Fn2 = F ′

n2 ∪ F ′′

n2. Then, Fn2 is a countable subset of Dn and F ′′

n2 isclosed in P(R(X)), since F ′

n1 is finite and and F ′′

n2 is closed in P(R(X)). Bythe definition of Fn2, we have

Xn2 ∪ (((∞ × [0, αn0]) × 0)× n) ⊆ St(Fn2,U).

Finally, we show that there exists a finite subset Fn of Dn such that Xn3 ⊆St(Fn3,U). Since ∞×κ+ is countably compact, then, By Lemma 2.1, A(∞×κ+) × n is acc and there exists a finite subset F ′

n3 ⊆ Dn2 such that

En = Xn3 \ St(F ′

n3,U) ⊆ ((∞ × κ+) × 0) × n is a finite subset

and each point of En is an isolated point of ((∞×κ+)×0)×n. For eachpoint x ∈ En, there exists a Ux ∈ U such that x ∈ Ux. For each point x ∈ En,pick dx ∈ Dn ∩ Ux. Let F ′′

n3 = dx : x ∈ E, then F ′′

n3 is a finite subset of Dn

and E ⊆ St(F ′′

n3,U). If we put Fn3 = F ′

n3 ∪ F ′′

n3, then Fn3 is a finite subset ofDn and

Xn3 ⊆ St(Fn3,U).

18 Y.-K. Song

If we put Fn = Fn1 ∪ Fn2 ∪ Fn3, then Fn is a countable subset of Dn suchthat

((X ′

n × 0)× ω) ∪ Xn ⊆ St(Fn,U).

Since Fn1 and Fn2 are closed in P(R(X)), Fn3 is finite and each point of Fn isisolated, then Fn is discrete closed in P(R(X)).

Let F = ∪n∈ωFn. Then, F is a countable subset of D and

St(F,U) = ∪n∈ωSt(Fn,U) ⊇ ∪n∈ω(((X ′

n × 0)× ω) ∪ Xn) = P(R(X)).

Since each point of F is isolated, then F is discrete in P(R(X)). Since Fn isdiscrete closed in Xn and Xn is open closed in P(R(X)) for each n ∈ ω, thenF has not accumulation points in R(X) × ω. On the other hand, since F iscountable and κ ≥ |X | > ω, then every point of Xω is not accumulation pointof F by the construction of the topology of P(R(X)). This shows that F isclosed in P(R(X)), which completes the proof.

Since every discretely absolutely star-Lindelof space is discretely star-Lindelof,the next corollary follows from Theorem 2.2.

Corollary 2.3. Every Hausdorff star-Lindelof space can be represented in a

Hausdorff discretely star-Lindelof space as a closed Gδ-subspace.

Since every discretely absolutely star-Lindelof space is absolutely star-Lindelof,the next corollary follows from Theorem 2.2.

Corollary 2.4. Every Hausdorff star-Lindelof space can be represented in a

Hausdorff absolutely star-Lindelof space as a closed Gδ-subspace.

The author [10] proved that every Hausdorff (regular, Tychonoff) linked-Lindelof space can be represented a closed Gδ-subspace in Hausdorff (regular,Tychonoff, respectively) star-Lindelof space. Thus, we have the next corollary.

Corollary 2.5. Every Hausdorff linked-Lindelof space can be represented in a

Hausdorff discretely absolutely star-Lindelof space as a closed Gδ-subspace.

On the separation of Theorem 2.2, Song [14] showed that R(X) is Tychonoffif X is a locally-countable (ie., each point of X has a neighborhood U with|U | ≤ ω) Tychonoff space. Thus, we have the following proposition by theconstruction of the topology of P(R(X)).

Proposition 2.6. If X is a locally countable Tychonoff space, then P(R(X))is Tychonoff.

By Theorem 2.2 and Proposition 2.6, we have the next corollary.

Corollary 2.7. Every locally-countable, star-Lindelof Tychonoff space can be

represented in a discretely absolutely star-Lindelof Tychonoff space as a closed

Gδ-subspace.

The author [10] proved that every Hausdorff (regular, Tychonoff) linked-Lindelof space can be represented a closed Gδ-subspace in Hausdorff (regular,Tychonoff, respectively) star-Lindelof space. Thus, we have the following corol-lary by Corollary 2.7.


Corollary 2.8. Every locally-countable, linked-Lindelof Tychonoff space can be

represented in a discretely absolutely star-Lindelof Tychonoff space as a closed

Gδ-subspace.

Remark 2.9. In Theorem 2.2, even if X is locally-countable normal, R(X)need not be normal (hence, P(R(X)) need not be normal). Indeed, X×0 andA(∞ × κ+) are disjoint closed subsets of R(X) that can not be separatedby disjoint open subsets of R(X). Thus, the author does not know if everylocally countable, normal star-Lindelof space can be represented in a normaldiscretely absolutely star-Lindelof space as a closed Gδ-subspace.

Remark 2.10. The author does not know if every regular (Tychonoff, nor-mal) star-Lindelof space can be represented in a regular (Tychonoff, normal,respectively) discretely absolutely star-Lindelof space as a closed subspace oras a closed Gδ-subspace.

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[12] Y.-K. Song, Regular closed subsets of absolutely star-Lindelof spaces, Questions Answers

Gen. Topology 22 (2004), 131–135.[13] Y.-K. Song, Some notes on star-Lindelof spaces, Questions Answers Gen. Topology 24

(2006), 11–15.[14] Y.-K. Song, Embedding into discretely absolutely star-Lindelof spaces, Comment. Math.

Univ. Carolinae 12 (2007), no. 2, 303–309.[15] J. E. Vaughan, Absolutely countably compactness and property (a), Talk at 1996 Praha

symposium on General Topology.[16] Y. Yasui and Z.-M. Gao, Spaces in countable web, Houston J. Math. 25 (1999), 327–335.

20 Y.-K. Song

Received November 2007

Accepted March 2009

Yan-Kui Song (songyankuinjnu.edu.cn)Institute of Mathematics, School of Mathematics and Computer Sciences, Nan-jing Normal University, Nanjing, 210097, P. R. China



Volume 10, No. 1, 2009

pp. 21-28

Best proximity pair theorems for relativelynonexpansive mappings

V. Sankar Raj and P. Veeramani∗

Abstract. Let A,B be nonempty closed bounded convex subsets ofa uniformly convex Banach space and T : A∪B → A∪B be a map suchthat T (A) ⊆ B, T (B) ⊆ A and ‖Tx − Ty‖ ≤ ‖x − y‖, for x in A andy in B. The fixed point equation Tx = x does not possess a solutionwhen A ∩ B = ∅. In such a situation it is natural to explore to findan element x0 in A satisfying ‖x0 − Tx0‖ = inf‖a − b‖ : a ∈ A, b ∈B = dist(A,B). Using Zorn’s lemma, Eldred et.al proved that sucha point x0 exists in a uniformly convex Banach space settings underthe conditions stated above. In this paper, by using a convergencetheorem we attempt to prove the existence of such a point x0 (calledbest proximity point) without invoking Zorn’s lemma.

2000 AMS Classification: 47H10

Keywords: best proximity pair, relatively nonexpansive map, cyclic contrac-tion map, strictly convex space, uniformly convex Banach space, fixed point,metric projection.

1. Introduction

Let A, B be nonempty subsets of a normed linear space (X, ‖ · ‖) and a mapT : A ∪ B → A ∪ B is said to be a relatively nonexpansive map if it satisfies(i) T (A) ⊆ B, T (B) ⊆ A and (ii) ‖Tx− Ty‖ ≤ ‖x− y‖, for all x ∈ A, y ∈ B.Note that a relatively nonexpansive map need not be continuous in general. Butif A ∩ B is nonempty, then the map T restricted to A ∩ B is a nonexpansiveself map. If the fixed point equation Tx = x does not possess a solution itis natural to explore to find an x0 ∈ A satisfying ‖x0 − Tx0‖ = dist(A, B) =inf‖a − b‖ : a ∈ A, b ∈ B. A point x0 ∈ A is said to be a best proximitypoint for T if it satisfies ‖x0 − Tx0‖ = dist(A, B).

∗Corresponding author.

22 V. Sankar Raj and P. Veeramani

In [2], Eldred et.al introduced a geometric concept called proximal normalstructure which generalizes the concept of normal structure introduced by Mil-man and Brodskii [7].

Definition 1.1 (Proximal normal structure [2]). A convex pair (K1, K2) ina Banach space is said to have proximal normal structure if for any closed,bounded, convex proximal pair (H1, H2) ⊆ (K1, K2) for which dist(H1, H2) =dist(K1, K2) and δ(H1, H2) > dist(H1, H2), there exists (x1, x2) ∈ H1 × H2

such that

δ(x1, H2) < δ(H1, H2), δ(x2, H1) < δ(H1, H2)

where δ(H1, H2) = sup ‖h1 − h2‖ : h1 ∈ H1, h2 ∈ H2.

Using the concept of proximal normal structure, Eldred et.al [2] proved theexistence of best proximity points for relatively nonexpansive mappings.

Theorem 1.2 ([2]). Let (A, B) be a nonempty, weakly compact convex pair ina Banach space (X, ‖ · ‖), and suppose (A, B) has proximal normal structure.Let T : A ∪ B → A ∪ B be a relatively nonexpansive map. Then there exists(x, y) ∈ A × B such that ‖x − Tx‖ = ‖Ty − y‖ = dist(A, B).

The proof of the above theorem invokes Zorn’s lemma and the proximalnormal structure idea. Also it has been proved that every closed boundedconvex pair (A, B) of a uniformly convex Banach space has proximal normalstructure and every compact convex pair has proximal normal structure.

In this paper, by using a convergence theorem we attempt to prove theexistence of a best proximity point without invoking Zorn’s lemma.

2. Preliminaries

In this section we give some basic definitions and concepts which are usefuland related to the context of our results. We shall say that a pair (A, B) ofsets in a Banach space satisfies a property if each of the sets A and B hasthat property. Thus (A, B) is said to be convex if both A and B are convex.(C, D) ⊆ (A, B) ⇔ C ⊆ A, D ⊆ B etc.

dist(A, B) = inf‖x − y‖ : x ∈ A, y ∈ B

A0 = x ∈ A : ‖x − y‖ = dist(A, B) for some y ∈ B

B0 = y ∈ B : ‖x − y‖ = dist(A, B) for some x ∈ A

Let X be a normed linear space and C be a nonempty subset of X. Then themetric projection operator PC : X → 2C is defined as

PC(x) = y ∈ C : ‖x − y‖ = dist(x, C), for each x ∈ X.

It is well known that the metric projection operator PC on a strictly convexBanach space X is a single valued map from X to C, where C is a nonemptyweakly compact convex subset of X .

In [6], Kirk et.al proved the following lemma which guarantees the nonempti-ness of A0 and B0.

Best proximity pair theorems for relatively nonexpansive mappings 23

Lemma 2.1 ([6]). Let X be a reflexive Banach space and A be a nonemptyclosed bounded convex subset of X, and B be a nonempty closed convex subsetof X. Then A0 and B0 are nonempty and satisfy PB(A0) ⊆ B0, PA(B0) ⊆ A0.

In [8], Sadiq Basha and Veeramani proved the following result.

Lemma 2.2 ([8]). If A and B are nonempty subsets of a normed linear spaceX such that dist(A, B) > 0, then A0 ⊆ ∂(A) and B0 ⊆ ∂(B) where ∂(C)denotes the boundary of C in X for any C ⊆ X.

Suppose (A, B) is a nonempty weakly compact convex pair of subsets in aBanach space X . Consider the map P : A ∪ B → A ∪ B defined as

P (x) =

PB(x), if x ∈ A

PA(x), if x ∈ B(2.1)

If X is a strictly convex Banach space, then the map P is a single valued mapand satisfies P (A) ⊆ B, P (B) ⊆ A.

Proposition 2.3. Let A, B be nonempty weakly compact convex subsets ofa strictly convex Banach space X. Let T : A ∪ B → A ∪ B be a relativelynonexpansive map and P : A∪B → A∪B be a map defined as in (2.1). ThenTP (x) = P (Tx), for all x ∈ A0 ∪ B0.

Proof. Let x ∈ A0. Then there exists y ∈ B such that ‖x−y‖ = dist(A, B). Bythe uniqueness of the metric projection on a strictly convex Banach space, wehave PB(x) = y, PA(y) = x. Since T is relatively nonexpansive, we have ‖Tx−Ty‖ ≤ ‖x − y‖ = dist(A, B). ie PA(Tx) = Ty. This implies that PA(Tx) =TPB(x)

This observation will play an important role in this article. In [3], Eldredand Veeramani introduced a notion of cyclic contraction and studied the exis-tence of best proximity point for such maps. We make use of the main resultsproved in [3] to obtain best proximity pair theorems for relatively nonexpansivemappings.

Definition 2.4 ([3]). Let A and B be nonempty subsets of a metric space X.A map T : A∪B → A∪B is said to be a cyclic contraction map if it satisfies :

(1) T (A) ⊆ B, T (B) ⊆ A(2) there exists k ∈ (0, 1) such that d(Tx, T y) ≤ kd(x, y)+(1−k)dist(A, B)

for each x ∈ A, y ∈ B

We can easily see that every cyclic contraction map satisfies d(Tx, T y) ≤d(x, y), for all x ∈ A, y ∈ B. In [3], a simple existence result for a best proximitypoint of a cyclic contraction map has been given. It states as follows:

Theorem 2.5 ([3]). Let A and B be nonempty closed subsets of a completemetric space X. Let T : A ∪ B → A ∪ B be a cyclic contraction map, let x0 ∈A and define xn+1 = Txn, n = 0, 1, 2, · · · . Suppose x2n has a convergentsubsequence in A. Then there exists x ∈ A such that d(x, Tx) = dist(A, B).


In uniformly convex Banach space settings, the following result proved in[3] ensures the existence, uniqueness and convergence of a best proximity pointfor a cyclic contraction map. We use this result to prove our main results.

Theorem 2.6 ([3]). Let A and B be nonempty closed and convex subsets ofa uniformly convex Banach space. Suppose T : A ∪ B → A ∪ B is a cycliccontraction map, then there exists a unique best proximity point x ∈ A( that iswith ‖x−Tx‖ = dist(A, B) ). Further, if x0 ∈ A and xn+1 = Txn, then x2nconverges to the best proximity point.

We need the notion of ”approximatively compact set” to prove a convergenceresult in the next section.

Definition 2.7 ([9]). Let X be a metric space. A subset C of X is said tobe approximatively compact if for any y ∈ X, and for any sequence xn in Csuch that d(xn, y) → dist(y, C) as n → ∞, then xn has a subsequence whichconverges to a point in C.

In a metric space, every approximatively compact set is closed and everycompact set is approximatively compact. Also a closed convex subset of auniformly convex Banach space is approximatively compact.

3. Main Results

The following convergence theorem will play an important role in our mainresults.

Theorem 3.1. Let X be a strictly convex Banach space and A be a nonemptyapproximatively compact convex subset of X and B be a nonempty closed sub-set of X. Let xn be a sequence in A and y ∈ B. Suppose ‖xn − y‖ →dist(A, B), then xn → PA(y).

Proof. Suppose that xn does not converges to PA(y), then there exists ε > 0and a subsequence xnk

of xn such that

‖xnk− PA(y)‖ ≥ ε(3.1)

Since xnk is a sequence in A such that ‖xnk

− y‖ → dist(A, B), and Ais approximatively compact, xnk

has a convergent subsequence xn′

k such

that xn′

k→ x for some x ∈ A. Then

‖xn′

k− y‖ → ‖x − y‖

also, ‖xn′

k− y‖ → dist(A, B) implies ‖x − y‖ = dist(A, B).

By the uniqueness of PA we have x = PA(y). But from (3.1) we have

ε ≤ ‖xn′

k− PA(y)‖ =⇒ 0 < ε ≤ ‖x − PA(y)‖ =⇒ x 6= PA(y)

which is a contradiction. Hence xn → PA(y).

The above theorem generalizes the following convergence result proved in [3]([3],Corollary 3.9) for a strictly convex Banach space.


Corollary 3.2 ([3]). Let A be a nonempty closed convex subset and B benonempty closed subset of a uniformly convex Banach space. Let xn be asequence in A and y0 ∈ B such that ‖xn−y0‖ → dist(A, B). Then xn convergesto PA(y0).

Remark 3.3. Let X be a normed linear space, let A be a nonempty closedconvex subset of X , and B be a nonempty approximatively compact convexsubset of X . If A0 is compact, then B0 is also compact.

Proof. If B0 is empty, then nothing to prove. Assume B0 is nonempty. Letyn be a sequence in B0. Then for each n ∈ N, there exists xn ∈ A0 suchthat ‖xn − yn‖ = dist(A, B). Since A0 is compact, there exists a convergentsubsequence xnk

which converges to some x ∈ A0. Consider the inequality,

‖ynk− x‖ ≤ ‖ynk

− xnk‖ + ‖xnk

− x‖ → dist(A, B).

Since B is approximatively compact, ynk has a convergent subsequence yn′

k

converges to some y ∈ B. Since B0 is closed, it implies that y ∈ B0. Hence B0

is compact.

Now we prove our main results.

Theorem 3.4. Let X be a uniformly convex Banach space. Let A be a nonemptyclosed bounded convex subset of X and B be a nonempty closed convex subsetof X. Let T : A ∪ B → A ∪ B be a relatively nonexpansive map. Then thereexist a sequence xn in A0 and x∗ ∈ A0 such that

(1) xn

w−→ x∗

(2) ‖x∗ − Tx∗‖ ≤ dist(A, B) + lim infn

‖Txn − Tx∗‖.

Proof. By Lemma 2.1, A0 is nonempty, hence there exist x0 ∈ A0 and y0 ∈ B0

such that ‖x0 − y0‖ = dist(A, B). For each n ∈ N, define a map Tn : A ∪ B →A ∪ B by

Tn(x) =

1

ny0 +

(

1 −1

n

)

Tx, if x ∈ A

1

nx0 +

(

1 −1

n

)

Tx, if x ∈ B

(3.2)

Since A and B are convex and T is a relatively nonexpansive map, for eachn ∈ N, Tn(A) ⊆ B, Tn(B) ⊆ A. Also for each x ∈ A, y ∈ B,

‖Tn(x) − Tn(y)‖ ≤1

n‖x0 − y0‖ +

(

1 −1

n

)

‖Tx − Ty‖

≤

(

1 −1

n

)

‖x − y‖ +1

ndist(A, B).(3.3)

This implies that for each n ∈ N, Tn is a cyclic contraction on A ∪ B. Henceby Theorem 2.6, for each n ∈ N there exists xn ∈ A such that

‖xn − Tnxn‖ = dist(A, B).(3.4)


Hence xn ∈ A0, for each n ∈ N. Since A0 is bounded, B0 is also bounded, andT (A0) ⊆ B0, Tn(A0) ⊆ B0. Also observe that for any x ∈ A0,

‖Tnx − Tx‖ ≤1

n‖y0 − Tx‖ ≤

1

nδ(B0) → 0 as n → ∞.(3.5)

Since A0 is a closed bounded convex set, xn has a weakly convergent sub-sequence. Without loss of generality, let us assume that xn itself weakly

converges to x∗, for some x∗ ∈ A0. Then xn − Tx∗w

−→ x∗ − Tx∗. Since ‖ · ‖ isweakly lower semi continuous, and by (3.4), (3.5) we have

‖x∗ − Tx∗‖ ≤ lim infn

‖xn − Tx∗‖

≤ lim infn

‖xn − Tnxn‖ + ‖Tnxn − Txn‖ + ‖Txn − Tx∗‖

≤ lim infn

dist(A, B) +1

nδ(B0) + ‖Txn − Tx∗‖

≤ dist(A, B) + lim infn

‖Txn − Tx∗‖

Hence the theorem.

We use the above theorem to prove :

Theorem 3.5. Let X be a uniformly convex Banach space. Let A be a nonemptyclosed bounded convex subset of X such that A0 is compact, and B be a nonemptyclosed convex subset of X. Let T : A∪B → A∪B be a relatively nonexpansivemap. Then there exist x∗ ∈ A such that ‖x∗ − Tx∗‖ = dist(A, B).

Proof. By Theorem 3.4, there exist a sequence xn in A0 and x∗ ∈ A0 such

that xn

w−→ x∗ and satisfies the inequality

‖x∗ − Tx∗‖ ≤ dist(A, B) + lim infn

‖Txn − Tx∗‖.

Since A0 is compact, xn converges to x∗ strongly. The proof will be completeif we show that ‖Txn − Tx∗‖ → 0.

Claim : ‖Txn − Tx∗‖ → 0 as n → ∞.It is enough to show that ‖Txn − PA(Tx∗)‖ → dist(A, B) as n → ∞. Then

by Theorem 3.1, we have Txn → PB(PA(Tx∗)) = Tx∗. Consider

‖xn − PBx∗‖ ≤ ‖xn − x∗‖ + ‖x∗ − PBx∗‖ → dist(A, B).

Since T is relatively nonexpansive we have,

‖Txn − PATx∗‖ = ‖Txn − T (PBx∗)‖ ≤ ‖xn − PBx∗‖ → dist(A, B).

This ends the claim and hence the theorem.

Theorem 3.6. Let X be a strictly convex Banach space, let A be a nonemptyclosed convex subset of X such that A0 is a nonempty compact set and B be anonempty closed convex subset of X. Let T : A ∪ B → A ∪ B be a relativelynonexpansive map. Then there exists x∗ ∈ A such that ‖x∗−Tx∗‖ = dist(A, B).


Proof. Since A0 is nonempty and compact, we can construct a sequence of cycliccontraction maps Tn : A ∪ B → A ∪ B as in Theorem 3.4. We use Theorem2.5 for an existence of best proximity point xn ∈ A0 such that ‖xn − Tnxn‖ =dist(A, B). Since A0 is compact, xn has a convergent subsequence xnk

such that xnk

→ x∗ for some x∗ ∈ A0. As in the proof of Theorem 3.5, we canshow Txnk

→ Tx∗. The proof ends by considering the following inequality,

‖x∗−Tx∗‖ ≤ ‖x∗−xnk‖+‖xnk

−Tnkxnk

‖+‖Tnkxnk

−Txnk‖+‖Txnk

−Tx∗‖

and by observing ‖Tnkxnk

− Txnk‖ ≤ 1

nk

δ(B0) → 0.

We give below some situations where A0 is a compact subset of A.

Example 3.7. Let A be a unit ball in a strictly convex Banach space X and Bbe a closed convex subset of X with dist(A, B) > 0. Then A0 contains atmostone point.

Proof. Clearly A0 is a bounded convex subset of A, moreover by Lemma 2.2,A0 is contained in the boundary of A. ie A0 ⊆ ∂A. Suppose x1, x2 ∈ A0 withx1 6= x2, then by strict convexity ‖x1+x2

2‖ < 1 which implies that x1+x2

2/∈ ∂A, a

contradiction to the convexity of A0. Hence A0 contains atmost one point.

Example 3.8. Let A be a nonempty closed bounded convex subset of a uni-formly convex Banach space X and B be a nonempty closed convex subset ofX such that span(B) is finite dimensional with dist(A, B) > 0. Then A0 andB0 are nonempty compact subsets of A, B respectively.

Proof. Let yn be a sequence in B0 then there exists a sequence xn inA0 such that ‖xn − yn‖ = dist(A, B). Since xn is bounded, yn is also abounded sequence in B0. Since B is finite dimensional, yn has a convergentsubsequence. Hence B0 is compact. Then by Remark 3.3, A0 is also a compactset.

Corollary 3.9. Let A be a nonempty closed bounded convex subset of a uni-formly convex Banach space X and B be a nonempty closed convex subset of Xsuch that span(B) is finite dimensional with dist(A, B) > 0. Let T : A ∪ B →A ∪ B be a relatively nonexpansive map. Then there exists x ∈ A such that‖x − Tx‖ = dist(A, B).

Acknowledgements. The authors would like to thank the referee for usefulcomments and suggestions for the improvement of the paper. The first authoracknowledges the Council of Scientific and Industrial Research(India) for thefinancial support provided in the form of a Junior Research Fellowship to carryout this research work.


References

[1] Handbook of metric fixed point theory, Edited by W. A. Kirk and Brailey Sims, KluwerAcad. Publ., Dordrecht, 2001. MR1904271 (2003b:47002)

[2] A. Anthony Eldred, W. A. Kirk and P. Veeramani, Proximal normal structure and

relatively nonexpansive mappings, Studia Math. 171, no. 3 (2005), 283–293.[3] A. Anthony Eldred and P. Veeramani, Existence and convergence of best proximity

points, J. Math. Anal. Appl. 323 (2006), 1001–1006.[4] M. A. Khamsi and W. A. Kirk, An introduction to metric spaces and fixed point theory,

Wiley-Interscience, New York, 2001. MR1818603 (2002b:46002)[5] W. A. Kirk, P. S. Srinivasan and P. Veeramani, Fixed points for mappings satisfying

cyclic contractive conditions, Fixed Point Theory 4, no. 1 (2003), 79–89.[6] W. A. Kirk, S. Reich and P. Veeramani, Proximal retracts and best proximity pair

theorems, Numer. Funct. Anal. Optim. 24 (2003), 851–862.[7] D. P. Milman and M. S. Brodskii, On the center of a convex set, Dokl. Akad. Nauk.

SSSR (N.S) 59 (1948), 837–840.[8] S. Sadiq Basha and P. Veeramani, Best proximity pair theorems for multifunctions with

open fiber, J. Approx. Theory. 103 (2000), 119–129. (2000)[9] S. Singh, B. Watson and P. Srivastava, Fixed point theory and best approximation: the

KKM-map principle, Kluwer Acad. Publ., Dordrecht, 1997. MR1483076 (99a:47087)

Received December 2007

Accepted January 2008

V. Sankar Raj (sankar [email protected])Department of Mathematics, Indian Institute of Technology Madras, Chennai600 036, India.

P. Veeramani ([email protected])Department of Mathematics, Indian Institute of Technology Madras, Chennai600 036, India.



Volume 10, No. 1, 2009

pp. 29-37

∗-half completeness in quasi-uniform spaces

Athanasios Andrikopoulos

Abstract. Romaguera and Sanchez-Granero (2003) have intro-duced the notion of T1

∗-half completion and used it to see when aquasi-uniform space has a ∗-compactification. In this paper, for anyquasi-uniform space, we construct a ∗-half completion, called stan-dard ∗-half completion. The constructed ∗-half completion coincideswith the usual uniform completion in the uniform spaces and is theunique (up to quasi-isomorphism) T1

∗-half completion of a symmetriz-able quasi-uniform space. Moreover, it constitutes a ∗-compactificationfor ∗-Cauchy bounded quasi-uniform spaces. Finally, we give an exam-ple which shows that the standard ∗-half completion differs from thebicompletion construction.

2000 AMS Classification: 54E15, 54D35.

Keywords: quasi-uniform, ∗-half completion, ∗-compactification.

1. Introduction and preliminaries

The problem of constructing compactifications of quasi-uniform spaces hasbeen investigated by several authors ([4, 3.47], [5], [7]). This notion of quasi-uniform compactification is by definition Hausdorff. Moreover, a point sym-metric totally bounded T1 quasi-uniform space may have many totally boundedcompactifications (see [5, page 34]) . Contrary to this notion, Romaguera andSanchez-Granero have introduced the notion of ∗-compactification of a T1 quasi-uniform space (see [8], [10] and [11]) and prove that: (a) Each T1 quasi-uniformspace having a T1

∗-compactification has an (up to quasi-isomorphism) uniqueT1

∗-compactification ([11, Corollary of Theorem 1]); and (b) All the Wallman-type compactifications of a T1 topological space can be characterized in termsof the ∗-compactification of its point symmetric totally transitive compatiblequasi-uniformities ([9, Theorem 1]). The proof of (a) is achieved with the helpof the notion of T1

∗-half completion of a quasi-uniform space, which is intro-duced in [11]. Following ([11, Theorem 1]), if a quasi-uniform space (X,U) is T1

30 A. Andrikopoulos

∗-half completable (it has a T1∗-half completion), then any T1

∗-half comple-tion of (X,U) is unique up to a quasi-isomorphism. In this paper, we prove thatevery quasi-uniform space has a ∗-half completion, called standard ∗-half com-pletion, which in the case of a uniform space coincides with the usual one. Wealso give an example which shows that the standard ∗-half completion and thebicompletion are in general different. While a quasi-uniform space may havemany ∗-half completions, here we prove that a symmetrizable quasi-uniformspace has an (up to a quasi-isomorphism) unique ∗-half completion. We alsoprove that the standard ∗-half completion constitutes a ∗-compactification for∗-Cauchy bounded quasi-uniform spaces.

Let us recall that a quasi-uniformity on a (nonempty) set X is a filter U onX × X such that for each U ∈ U , (i) ∆(X) = (x, x)|x ∈ X ⊆ U , and (ii)V V ⊆ U for some V ∈ U , where V V = (x, y) ∈ X × X | there is z ∈ Xsuch that (x, z) ∈ V and (z, y) ∈ V . The pair (X,U) is called a quasi-uniformspace. If U is a quasi-uniformity on a set X , then U−1 = U−1|U ∈ Uis also a quasi-uniformity on X called the conjugate of U . Given a quasi-uniformity U on X , U⋆ = U

∨U−1 will denote the coarsest uniformity on

X which is finer than U . If U ∈ U , the entourage U ∩ U−1 of U⋆ will bedenoted by U⋆. The topology τ(U) induced by the quasi-uniformity U on Xis G ⊆ X | for each x ∈ G there is U ∈ U such that U(x) ⊆ G whereU(x) = y ∈ X |(x, y) ∈ U. If (X, τ) is a topological space and U is a quasi-uniformity on X such that τ = τ(U) we say that U is compatible with τ . Let(X,U) and (Y,V) be two quasi-uniform spaces. A mapping f : (X,U) → (Y,V)is said to be quasi-uniformly continuous if for each V ∈ V there is U ∈ U suchthat (f(x), f(y)) ∈ V whenever (x, y) ∈ U . A bijection f : (X,U) → (Y,V)is called a quasi-isomorphism if f and f−1 are quasi-uniformly continuous. Inthis case we say that (X,U) and (Y,V) are quasi-isomorphic. A filter B is calledU⋆-Cauchy if and only if for each U ∈ U there exists B ∈ B such that B×B ⊆ U(see [4, page 48]). A net (xa)a∈A is called U⋆-Cauchy net if for each U ∈ Uthere exists an a

U∈ A such that (xa, x

a′ ) ∈ U whenever a ≥ aU, a′ ≥ a

U.

We call aU

extreme index of (xa)a∈A for U and xa

Uextreme point of (xa)a∈A

for U . A quasi-uniform space (X,U) is half complete if every U⋆-Cauchy filteris τ(U)-convergent [2]. Following to [11, Theorem 1], a ∗-half completion of aT1 quasi-uniform space (X,U) is a half complete T1 quasi-uniform space (Y,V)that has a τ(V⋆)-dense subspace quasi-isomorphic to (X,U). In [11, Definition3] also the authors introduce and study the notion of a ∗-compactification a T1

quasi-uniform space. A ∗-compactification of a T1 quasi-uniform space (X,U)is a compact T1 quasi-uniform space (Y,V) that has a τ(V⋆)-dense subspacequasi-isomorphic to (X,U).

2. The ∗-half-completion

According to Doitchinov [3, Definition 1], a net (yβ)

β∈Bis called a conet of

the net (xa)a∈A, if for any U ∈ U there are aU

∈ A and βU

∈ B such that

∗-half completeness in quasi-uniform spaces 31

(yβ, xa) ∈ U whenever a ≥ aU

and β ≥ βU. In this case, we write (y

β, x

a) −→ 0.

We denote (x) the constant net (xa)

a∈A, for which x

a= x for each a ∈ A.

Definition 2.1 (see [1, Definitions 1.1(3)]). Let (X,U) be a quasi-uniform space.

(1) For every U⋆-Cauchy net (xa)

a∈Awe consider a U⋆-Cauchy net (y

β)

β∈B

which is a conet of (xa)

a∈A, different than (x

a)

a∈A. In the following,

we consider all the nets A = (xia)

a∈Ai|i ∈ I that have (y

β)

β∈Bas

their conet including (yβ)

β∈Bitself. In the next, we pick up all the nets

B = (yj

β)

β∈Bj|j ∈ J which are conets of all the elements of A. The

ordered couple (A,B) have the following properties:

(a) for every U ∈ U and every (xai)a∈Ai

∈ A, (yβ

j

)β∈Bj

∈ B there

are indices aU

i, βU

j

such that (yβ

j

, xia) ∈ U whenever a ≥ a

U

i and

β ≥ βU

j .

We call aU

i (resp. βU

j

) extreme index of (xai)a∈Ai

(resp. (yβ

j

)β∈Bj

)

for U and xia

Ui (resp. y

j

βU

j ) extreme point of (xai)a∈Ai

(resp. (yβ

j

)β∈Bj

)

for U .(b) B contains all the conets of all the elements of A and conversely

A contains all the nets whose conets are all the elements of B.We call the ordered pair (A,B) h∗-cut, the nets (xa)a∈A and (y

β)

β∈B

generator and co-generator of (A,B) respectively. We also say thatthe pair ((y

β)

β∈B, (xa)a∈A) generates the h∗-cut (A,B). It is clear

that different pairs of U⋆-Cauchy nets can generate the same h∗-cut.The families A and B are called classes (first and second respec-

tively) of the h∗-cut (A,B). In the following, X denotes the set ofall h∗-cuts in X.

If the above U⋆-Cauchy net (xa)

a∈Ahas not as conet a U⋆-Cauchy net

different from itself, then we relate to it the h∗-cut which generated bythe pair ((xa)a∈A, (xa)a∈A).

(2) To every x ∈ X we assign an h∗-cut, denoted φ(x) = (Aφ(x)

,Bφ(x)

),

which is generated by the pair ((x), (x)). Clearly, x belongs to bothof A

φ(x)and B

φ(x). Thus the class A

φ(x)contains all the nets which

converge to x in τU

and Bφ(x)

contains nets which converge to x inτU−1

.(3) Suppose that K = (x

a)

a∈A|(x

a)

a∈Ais a non τ(U)-convergent U⋆-Cauchy

net. Let XK

= ξ ∈ X| the generator of ξ belongs to K. Then we

put X = φ(X) ∪ XK

.(4) We often say for a U⋆-Cauchy net (xa)a∈A with a conet (y

β)β∈B and

U ∈ U that:

“finally ((yβ)

β, (xa)a) ∈ U”or in symbols “τ.((y

β)

β, (xa)a) ∈ U ”,

if there are aU

∈ A and βU

∈ B such that (yβ, xa) ∈ U whenever

a ≥ aU, β ≥ β

U.

32 A. Andrikopoulos

Definition 2.2. Let (X,U) be a quasi-uniform space, ξ ∈ X and W ∈ U .

(1) We say that a net (tγ)

γ∈Γis W -close to ξ, if for each net (xi

a)a∈Ai∈ A

ξ

there holds τ.((tγ)

γ, (xi

a)a) ∈ W .

(2) For each U ∈ U denote by U the collection of all pairs (ξ′, ξ′′) for whicha co-generator of ξ′ is U -close to ξ′′.

The proof of the following result is straightforward, so it is omitted.

Proposition 2.3. Let (X,U) be a quasi-uniform space and let (yβ)

β∈Bbe a

co-generator of an h∗-cut ξ in X. Then (yβ)

β∈Bbelongs to both of the classes

Aξ and Bξ.

As an immediate consequence of Definition 2.2 and Proposition 2.3 we obtainthe following proposition.

Proposition 2.4. Let (X,U) be a quasi-uniform space, ξ′, ξ′′ ∈ X and U ∈ U .If (y

β)

β∈B, (y

γ)

γ∈Γare co-generators of ξ′ and ξ′′ respectively, then

(ξ′, ξ′′) ∈ U if and only if τ.((yβ)

β, (y

γ)

γ) ∈ U .

Corollary 2.5. Let (X,U) be a quasi-uniform space and let ξ′, ξ′′ ∈ X. If(y

β)

β∈B, (y

γ)

γ∈Γare co-generators of ξ′ and ξ′′ respectively, then

ξ′ = ξ′′ if and only if (yβ, y

γ) −→ 0 in τ(U⋆).

The following lemma is obvious.

Lemma 2.6. Let U, V ∈ U . Then U ⊆ V if and only if V ⊆ U .

Theorem 2.7. The family U = U |U ∈ U is a base for a quasi-uniformity Uon X.

Proof. By definitions 2.2 and Proposition 2.3, it follows that the pair (ξ, ξ)belongs to every element of U and by the previous Lemma U is a filter.

Let now U, W ∈ U be such that W W W ⊆ U and x, y ∈ X with(x, y) ∈ W W . Then there exists a z in X such that (x, z) ∈ W and (z, y) ∈ W .If (xx

a )a∈A

, (zzγ)

γ∈Γ and (yyβ)

β∈Bare co-generators of x, z and y respectively,

then Definition 2.2 and Proposition 2.3 imply that τ.((xxa )a, (zz

γ)

γ) ∈ W and

τ.((zzγ)γ , (yy

β)

β) ∈ W . We note that, for each (t

δ)

δ∈∆∈ A

y, it holds that

τ.(yyβ, t

δ) −→ 0. Hence, τ.((xx

a )a, (tδ)

δ) ∈ W W W ⊆ U which implies that

(x, y) ∈ U .

Proposition 2.8. If ξ ∈ X and (xa)a∈A is a U⋆-Cauchy net which belong toA

ξ, then φ(xa) −→ ξ. Dually, if (y

β)

β∈Bis a U⋆-Cauchy net which belong to

Bξ, then lim

β(φ(y

β), ξ) = 0.

Proof. Let V , U ∈ U such that V V ⊆ U . If (zγ)

γ∈Γis a co-generator of

ξ, then (zγ, x

a) −→ 0. Thus there are a

Vand γ

Vsuch that (z

γ, x

a) ∈ V for

γ ≥ γV

and a ≥ aV. Fix an a ≥ a

Vand pick a net (x

δ)

δ∈∆of A

φ(xa ). Then,

xδ−→ xa and so (xa, x

δ) ∈ V , whenever δ ≥ δ

Vfor some δ

V∈ ∆. Hence,

(zγ, x

δ) ∈ U for γ ≥ γ

Vand δ ≥ δ

V. Hence (ξ, φ(xa)) ∈ U , whenever a ≥ a

V.


The proof of the dual is similar.

Theorem 2.9. The quasi-uniform space (X,U) is a ∗-half completion of (X,U).

Proof. We firstly prove that (X,U) is half-complete, and secondly that the

space (X,U) has a τ(U⋆)-dense subspace quasi-isomorphic to (X,U). Indeed,

let (ξa)a∈A be a U⋆-Cauchy net of X . In the following, for each a ∈ A, (ya

β)

β∈Ba

denotes a co-generator of ξa. Suppose that W ∈ U . Then, there exists a

W∈ A

such that (ξγ , ξa) ∈ W whenever γ, a ≥ aW

. Fix an a ≥ aW

and suppose thatβ(a, W ) is the extreme index of (ya

β)

β∈Bafor W .

We consider the setA⋆ = (a, W )|a ∈ A, W ∈ U

ordered by (a′, W ′) ≤ (a′′, W ′′) if a′ ≤ a′′ and W ′′ ⊆ W ′.We put y(a, W ) = ya

β(a,W )and we prove that the net

y(a, W )|(a, W ) ∈ A⋆

is a U⋆-Cauchy net.Indeed, let U ∈ U . Pick V ∈ U such that V V V ⊆ U . Suppose

that (a′, W ′), (a′′, W ′′) ≥ (aV, V ). Then, (y(a′, W ′), ya′

β′) ∈ (W ′)⋆ ⊆ V ⋆ and

(y(a′′, W ′′), ya′′

β′′) ∈ (W ′′)⋆ ⊆ V ⋆ whenever β′ ≥ β′(a′, W ′) and β′′ ≥ β′′(a′′, W ′′).

Since (ξa)a∈A is a U⋆-Cauchy net of X, Proposition 2.4 implies that

τ.((ya′

β′)

β′ , (ya′′

β′′)

β′′ ) ∈ V ⋆ whenever a′, a′′ ≥ aV. Hence, (y(a′, W ′), y(a′′, W ′′)) ∈

V ⋆ V ⋆ V ⋆ ⊆ U⋆.

We now prove that (ξa)a∈A is τ(U)-convergent. We have two cases.

Case 1. (y(a, W ))(a,W )∈A⋆ τ(U)-converges to a point x ∈ X.

In this case, we have that (φ(y(a, W )))(a,W )∈A⋆ τ(U)-converges to φ(x). Since

(yaβ)

β∈Babelongs to B

ξa, Proposition 2.8 implies that (φ(y(a, W )), ξ

a) −→ 0.

Hence, from (φ(x), φ(y(a, W ))) −→ 0 we conclude that (ξa)a∈A τ(U)-convergesto φ(x).

Case 2. (y(a, W ))(a,W )∈A⋆ is a non τ(U)-convergent net.

Let ξ be the h∗-cut in X which is generated by (y(a, W ))(a,W )∈A⋆ . It follows,

by Proposition 2.8, that (ξ, φ(y(a, W ))) → 0. Since (yaβ)

β∈Babelongs to B

ξa,

Proposition 2.8 implies that (φ(y(a, W )), ξa) −→ 0. The rest is obvious.

It remains to prove that (φ(X),U/φ(X) × φ(X)) is a τ(U⋆)-dense subspace

of (X,U). Indeed, let ξ ∈ X and let (yβ)

β∈Bbe a co-generator of it. Then,

since the co-generator belongs to both of classes of ξ, Proposition 2.8 implies

that φ(yβ) τ(U

⋆)-converges to ξ.

In the sequel the ∗-half completion (X,U) constructed above will be calledstandard ∗-half completion of the space (X,U).

The following example shows that the standard ∗-half completion and thebicompletion of a quasi-uniform space are in general different.

34 A. Andrikopoulos

Example 2.10. Let X be the set consisting of all nonzero real numbers andlet d be the quasi-metric on X given by:

d(x, y) =

y − x if x < y

0 otherwise

Suppose that U is the quasi-uniformity generated by d. Let F be the U⋆-Cauchyfilter generated by (0, a)|a > 0 and G be the U⋆-Cauchy filter generated by(b, 0)|b < 0. Then a new point is defined by the h∗-cut ξ = (A

ξ,B

ξ), where

Aξ

= G,F and Bξ

= F. Hence, X = φ(X) ∪ ξ. Clearly, ξ defines the

point 0 in (X,U). On the other hand, there is exactly one minimal U⋆-Cauchyfilter coarser than F and G respectively. More precisely, if F

0and G

0are any

bases for F and G respectively, then U(F0) |F

0∈ F

0and U is a symmetric

member of U⋆ and U(G0) |G

0∈ G

0and U is a symmetric member of U⋆

are equivalent bases for the minimal U⋆-Cauchy filter H coarser than F and G

respectively. Hence, we have X = i(X) ∪ H. The filter H defines the point

0 in (X, U) as well. We conclude the following:

(i) The bicompletion of (X,U) differs from the standard ∗-half completion.Indeed, by the definition of ξ and from the Propositions 2.3 and 2.8,we conclude that φ(G) and φ(F) converge to 0 with respect to τ(U)

and τ(U⋆

) respectively. On the other hand, i(G) and i(F) converge to

0 with respect to τ(U⋆

).(ii) The standard ∗-half completion is not quasi-uniformly isomorphic to its

bicompletion. This is true by (i) and the fact that the bicompletion of(X,U) coincides up to a quasi-isomorphism with the bicompletion of(X,U).

Theorem 2.11. Let (X,U) be a uniform space. Then, the standard ∗-halfcompletion (X,U) coincides with the usual uniform completion.

Proof. Let (X,U) be a uniform space and let ξ be an h∗-cut in X . Supposethat (xa)a∈A ∈ A

ξand (y

β)

β∈B∈ B

ξ. Then (y

β, xa) −→ 0 and (xa, y

β) −→ 0.

Hence the nets and the conets of ξ coincide. Thus, the class of equivalentCauchy nets, of the uniform case, is identified with an h∗-cut and vice versa.Hence the “ground set”of the two completions is the X. The rest is obvious.

Next, we give an equivalent definition for nets for the Definition 5 in [11].

Definition 2.12. Let (X,U) be a quasi-uniform space. A U⋆-Cauchy net(x

a)

a∈Aon X is said to be symmetrizable if whenever (y

β)

β∈Bis a U⋆-Cauchy

net on X such that (yβ, x

a) −→ 0, then (x

a, y

β) −→ 0.

Definition 2.13. A quasi-uniform space (X,U) is called symmetrizable if eachU⋆-Cauchy net on X, including for each x ∈ X the constant net (x), is sym-metrizable.

It easy to check that a quasi-uniform space is symmetrizable if and only if thebicompletion is T

1. In this case, the space has only one T

0∗-half completion,


the bicompletion. From Theorem 2.9 and [11, Theorem 1] we immediate deducethe following result.

Corollary 2.14. If a T1 quasi-uniform space is symmetrizable, then it has aT1

∗-half completion which is unique up to a quasi-isomorphism.

3. Standard ∗-half completion and ∗-Compactification

We recall some well known notions from [6].A net (x

a)

a∈Ais said to be frequently in S, for some subset S of X , if and

only if for all a ∈ A there is some a′ ≥ a such that xa′ ∈ S. A net is said to

be eventually in S if and only if there is an a0

in A such that for all a ≥ a0,

xa

is in S. A point x in X is a cluster point of the net (xa)

a∈Aif and only if

the net is frequently in all neighborhoods of x. The net (xa)

a∈Aconverges to

x if and only if (xa)

a∈Ais eventually in all neighborhoods of x. The tail sets

of (xa)

a∈Aare the sets Ta (a in A) where Ta = x

a′ |a′ ≥ a. Note that the Ta

have the finite intersection property, by the directedness of the index set A, sothey generate a filter, the filter of tails of (x

a)

a∈Aor the filter associated with

the net (xa)

a∈A. Then a point x is a cluster point of (x

a)

a∈Aif and only if x is

in cl(Ta) for all a (if and only if it is a cluster point of the filter of tails). Andx

a−→ x if and only if the filter of tails converges to x. This already shows that

there is a close relationship between convergence of filters and convergence ofnets.

Definition 3.1 (see [6, page 81]). A universal net in X is one such that foreach S ⊂ X, either the net is eventually in S, or it is eventually in X \ S.

From the classical theory we have the following statements.

(a) A net is a universal net if and only if its associated filter is an ultrafilter.(b) Let F be the filter associated with the net (x

a)

a∈Aand G be a filter

with F ⊂ G. Then (xa)

a∈Ahas a subnet whose associated filter is G.

(a) and (b) implies that:

(c) Every net has a universal subnet.(d) A universal net converges to each of its cluster points.(e) A space is compact if and only if every universal net is convergent.

Definition 3.2 (see [11, Definition 6]). A quasi-uniform space (X,U) is called⋆-Cauchy bounded if for each ultrafilter F on X there is a U⋆-Cauchy filter Gon X such that (G,F) −→ 0.

Definition 3.2 admits an equivalent definition for nets.

Definition 3.3. A quasi-uniform space (X,U) is called ⋆-Cauchy bounded iffor each universal net (x

a)

a∈Aon X there is a U⋆-Cauchy net (y

β)

β∈Bon X

such that (yβ, x

a) −→ 0.

Theorem 3.4. Let (X,U) be a ⋆-Cauchy bounded quasi-uniform space. Thenthe standard ⋆-half completion (X,U) is a ⋆-compactification of the space (X,U).

36 A. Andrikopoulos

Proof. Let (ξa)

a∈Abe a universal net in (X,U). Suppose that for any a ∈

A, ξa

= (Aξa

,Bξa

). Let (yaβ)

β∈Baand y(a, W )|(a, W ) ∈ A⋆ be as in the

proof of Theorem 2.9. Then, y(a, W )|(a, W ) ∈ A⋆ is a net in X . By theabove statement (c), we have that (y(a, W ))

(a,W )∈A⋆ has a universal subnet,

let y(ak, W

k)|(a

k, W

k) ∈ A⋆, k ∈ K. Since (X,U) is ⋆-Cauchy bounded,

there is a U⋆-Cauchy net (xγ)

γ∈Γof X such that (x

γ, y(a

k, W

k)) −→ 0. Hence

(φ(xγ), φ(y(a

k, W

k))) −→ 0 in (X,U) (1). On the other hand, since the space

(X,U) is half-complete, there exists ξ ∈ X such that (φ(xγ))

γ∈Γτ(U)-converges

to ξ (2). Hence by (1) and (2) we conclude that φ(y(ak, W

k))|(a

k, W

k) ∈

A⋆, k ∈ K τ(U)-converges to ξ. Since φ(y(ak, W

k))|(a

k, W

k) ∈ A⋆, k ∈ K is

a subnet of φ(y(a, W ))(a,W )∈A⋆ we conclude that ξ is a cluster point of the latter.

Since (yaβ)

β∈Babelongs to B

ξa, Proposition 8 implies that (φ(y(a, W )), ξ

a) −→

0. Hence, ξ is a cluster point of (ξa)

a∈A. There also holds that (ξ

a)

a∈Ais a

universal net, thus the above statement (d) implies that it τ(U )-converges toξ. Finally, by the above statement (e) we conclude that the space (X,U) is

compact. By Theorem 9, the space (X,U) has a τ(U⋆)-dense subspace quasi-

isomorphic to (X,U). Hence (X,U) is a ⋆-compactification of (X,U).

References

[1] A. Andrikopoulos, Completeness in quasi-uniform spaces, Acta Math. Hungar. 105

(2004), 549-565, MR 2005f:54050.[2] J. Deak, On the coincidence of some notions of quasi-uniform completeness defined by

filter pairs, Stud. Sci. Math. Hungar. 26 (1991), 411-413, MR 94e:94077.[3] D. Doitchinov, A concept of completeness of quasi-uniform spaces, Topology Appl. 38

(1991), 205-217, MR 92b:54061.[4] P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Lectures Notes in Pure and

Appl. Math. 77 (1978), Marc. Dekker, New York, MR 84h:54026.

[5] P. Fletcher, and W. F. Lindgren, Compactifications of totally bounded quasi-uniform

spaces, Glasgow Math. J. 28 (1986), 31-36, MR 87f:54037.[6] J. Kelley, General Topology, D.Van Nostrand Company, Inc., Toronto-New York-

London, (1955), MR 16, 1136c.[7] H. Render, Nonstandard methods of completing quasi-uniform spaces, Topology Appl.

62 (1995), 101-125, MR 96a:54041.[8] S. Romaguera and M. A. Sanchez-Granero, *-Compactifications of quasi-uniform paces,

Stud. Sci. Math. Hung. 44 (2007), 307-316.[9] S. Romaguera and M. A. Sanchez-Granero, A quasi-uniform characterization of

Wallman type compactifications, Stud. Sci. Math. Hung. 40 (2003), 257-267, MR2004h:54021.

[10] S. Romaguera and M. A. Sanchez-Granero, Compactifications of quasi-uniform hyper-

spaces, Topology Appl. 127 (2003), 409-423, MR 2003j:54011.[11] S. Romaguera and M. A. Sanchez-Granero, Completions and compactifications of quasi-

uniform spaces, Topology Appl. 123 (2002), 363-382, MR 2003c:54051.


Received January 2008

Accepted August 2008

Athanasios Andrikopoulos ([email protected])Department of Economics, University of Ioannina, Greece



Volume 10, No. 1, 2009

pp. 39-48

Condensations of Cp(X) onto σ-compact spaces

V. V. Tkachuk∗

Abstract. We show, in particular, that if nw(Nt) ≤ κ for any t ∈ T

and C is a dense subspace of the product∏Nt : t ∈ T then, for

any continuous (not necessarily surjective) map ϕ : C → K of C into acompact space K with t(K) ≤ κ, we have Ψ(ϕ(C)) ≤ κ. This result hasseveral applications in Cp-theory. We prove, among other things, thatif K is a non-metrizable Corson compact space then Cp(K) cannotbe condensed onto a σ-compact space. This answers two questionspublished by Arhangel’skii and Pavlov.

2000 AMS Classification: Primary 54H11, 54C10, 22A05, 54D06; Secondary54D25, 54C25

Keywords: condensation, continuous image, Lindelof Σ-space, σ-compactspace, topology of pointwise convergence, network weight, tightness, Lindelofspace.

1. Introduction.

A weaker topology on a space X can be considered an approximation of thetopology of X . If this approximation has some nice properties then we canobtain a lot of useful information about the space X . Thus it is natural to findout when a space has a weaker compact topology. This is an old topic and anextensive research has been done here both in general topology and descriptiveset theory.

The quest for nice condensations of function spaces had its origin in func-tional analysis after Banach asked whether every separable Banach space hasa weaker compact metrizable topology. This problem was solved positively byPytkeev [9]. Answering a question of Arhangel’skii, Casarrubias–Segura showedin [5] that function spaces of Cantor cubes have a weaker Lindelof topology butit is consistent that some of them do not have a weaker compact topology.

∗Research supported by Consejo Nacional de Ciencia y Tecnologıa (CONACYT) deMexico, grant 400200-5-38164-E

40 V. V. Tkachuk

Arhangel’skii and Pavlov [4] studied systematically whenCp(X) has a weakercompact topology and formulated some open questions on weaker σ-compacttopologies on Cp(X). It is also worth mentioning that Marciszewski [7] gave aconsistent example of a space X ⊂ R such that Cp(X) does not have a weakerσ-compact topology.

In this paper we consider product spaces N =∏

t∈T Nt such that nw(Nt) ≤κ for all t ∈ T . We prove that if C is a dense subspace of N and ϕ : C → K

is a continuous (not necessarily surjective) map of C into a compact space Kwith t(K) ≤ κ then Ψ(ϕ(C)) ≤ κ, i.e., every closed subset of ϕ(C) is theintersection of at most κ-many open subsets of ϕ(C). This result has severalimportant applications in Cp-theory. We establish, in particular, that if Xis an ω-monolithic space such that l(Cp(X)) = t(Cp(X)) = ω and Cp(X)condenses onto a σ-compact space then X is cosmic. As a consequence, ifX is a non-metrizable Corson compact space then Cp(X) does not condenseonto a σ-compact space. This answers Questions 29 and 30 of the paper ofArhangel’skii and Pavlov [4].

Any compact space of countable tightness has countable π-character (see [1,Theorem 2.2.20]). This easily implies that if Cp(X) embeds in such a compactspace then X is countable. Therefore it is natural to conjecture that everycontinuous image of Cp(X) has a countable network whenever it embeds ina compact space of countable tightness. Another reason to believe that thisconjecture might be true is a theorem of Tkachenko [12] which states that ifa compact space K of countable tightness is a continuous image of a LindelofΣ-group then K is metrizable. At the present moment nothing contradicts thehypothesis that if G is a Lindelof Σ-group and ϕ : G→ K is a continuous map,where K is compact and t(K) ≤ ω then ϕ(G) has a countable network. Weprove this conjecture for the spaces Cp(X) with the Lindelof Σ-property.

2. Notation and terminology.

All spaces under consideration are assumed to be Tychonoff. If X is a spacethen τ(X) is its topology and τ∗(X) = τ(X) \ ∅; given an arbitrary setA ⊂ X let τ(A,X) = U ∈ τ(X) : A ⊂ U. If x ∈ X then we write τ(x,X)instead of τ(x, X). The space R is the real line with its natural topology andN = ω \ 0. If X and Y are spaces then Cp(X,Y ) is the space of real-valuedcontinuous functions from X to Y endowed with the topology of pointwiseconvergence. We write Cp(X) instead of Cp(X,R). The expression X ≃ Y

says that the spaces X and Y are homeomorphic.A family N of subsets of a space Z is called a network if for any U ∈ τ(Z)

there is N ′ ⊂ N such that⋃N ′ = U . The network weight nw(Z) of a space

Z is the minimal cardinality of a network in Z. A space X is called cosmic ifthe network weight of X is countable. If x ∈ X then a family A is a networkof X at the point x if x ∈

⋂A and for any U ∈ τ(x,X) there is A ∈ A such

that A ⊂ U .A family B ⊂ τ∗(X) is a π-base of X at a point x ∈ X if for any U ∈ τ(x,X)

there is B ∈ B with B ⊂ U . The minimal cardinality of a π-base of X at x is

Condensations of Cp(X) onto σ-compact spaces 41

denoted by πχ(x,X) and πχ(X) = supπχ(x,X) : x ∈ X. If ϕ is a cardinalinvariant then hϕ(X) = supϕ(Y ) : Y ⊂ X is the hereditary version of ϕ. IfXis a space and F is a closed subset of X then pseudocharacter ψ(F,X) of the setF in the space X is the minimal cardinality of a family U ⊂ τ(F,X) such that⋂U = F ; let ψ(X) = supψ(x, X) : x ∈ X and Ψ(X) = supψ(F,X) : F is

a closed subset of X.The tightness t(X) of a space X is the minimal cardinal κ such that, for any

A ⊂ X , if x ∈ A then there is B ⊂ A with |B| ≤ κ such that x ∈ B. We usethe Russian term condensation to denote a continuous bijection. A space Z iscalled κ-monolithic if for any A ⊂ Z with |A| ≤ κ, we have nw(A) ≤ κ.

If we have a product Z =∏

t∈T Zt and A ⊂ T then ZA =∏

t∈A Zt isthe A-face of Z and πA : Z → ZA is the natural projection. A set F ⊂ Z

depends on A ⊂ T if π−1A πA(F ) = F ; if F depends on a set of cardinality ≤ κ

then we say that F depends on at most κ-many coordinates. A set E ⊂ Z

covers a face ZA if πA(E) = ZA. Suppose that, for every t ∈ T we have afamily Nt of subsets of Zt and let N = Nt : t ∈ T . If we have a faithfullyindexed set A = t1, . . . , tn ⊂ T and Ni ∈ Nti

for each t ≤ n then let[t1, . . . , tn, N1, . . . , Nn] = x ∈ Z : x(ti) ∈ Ni for all i = 1, . . . , n. A set H ⊂ Z

is called N -standard (or standard if N is clear) if H = [t1, . . . , tn, N1, . . . , Nn]for some t1, . . . , tn ∈ T and Ni ∈ Nti

for all i ≤ n. In this case we letsupp(H) = A and r(H) = n. We also consider that H = Z is the uniquestandard subset of Z such that r(H) = 0. Given any point x ∈ Z and A ⊂ T

the set 〈x,A〉 = y ∈ Z : y(t) = x(t) for any t ∈ A is closed in Z. If A ⊂ T

then the face ZA is called κ-residual if |T \A| ≤ κ. Say that a non-empty closedset F ⊂ K is κ-large if, for any x ∈ F and any finite A ⊂ T , the set 〈x,A〉 ∩ Fcovers a κ-residual face of K.

All other notions are standard and can be found in [6] and [3].

3. Nice continuous images of function spaces.

Our results will be obtained by strengthening a result of Shirokov [11]. Al-though our modifications of Shirokov’s method are minimal, we give completeproofs because the paper [11] has never been translated and, even in Russian,it is completely out of access for a Western reader. In particular, we presentthe proof of the following lemma established in [11].

Lemma 3.1. Given an infinite cardinal κ suppose that nw(Nt) ≤ κ for anyt ∈ T and N =

∏t∈T Nt. Assume that C ⊂ N is dense in N , and we have a

compact extension Kt of the space Nt for any t ∈ T . If a set F ⊂ K =∏

t∈T Kt

is κ-large then there exists a Gκ-set G in the space K such that F ⊂ G andF ∩ C = G ∩ C. In particular, F ∩ C is a Gκ-subset of C.

Proof. We can assume, without loss of generality, that K \ F 6= ∅. For everyt ∈ T fix a network Nt in the space Nt such that |Nt| ≤ κ; we will need thefamily Mt = clKt

(N) : N ∈ Nt. If M = Mt : t ∈ T then the M-standardsubsets of K will be called standard. It is easy to see that

(1) the family H of all standard subsets of K is a network in K at every x ∈ C.

42 V. V. Tkachuk

Given standard sets P and P ′ say that P ′ P if P = [t1, . . . , tn,M1, . . . ,Mn]and there exists a natural k ≤ n such that P ′ = [ti1 , . . . , tik

,Mi1 , . . . ,Mik] for

some distinct i1, . . . , ik ∈ 1, . . . , n; if k < n then we write P ′ ≺ P . We alsoinclude here the case when k = 0 so P ′ = K P for any standard set P .Say that a standard set P is minimal if P ∩ F = ∅ but P ′ ∩ F 6= ∅ wheneverP ′ ≺ P . It follows from (1) that

(2) for any x ∈ C \ F there exists a minimal standard set P such that x ∈ P .

It will be easy to finish our proof if we establish that

(3) the family S of minimal standard sets has cardinality not exceeding κ.

Assume, toward a contradiction that |S| > κ. Then we can choose S0 ⊂ Ssuch that |S0| = κ+ and there exists n ∈ ω with r(P ) = n for all P ∈ S0.Observe first that

(4) if A ⊂ T , a set D ⊂ K covers the face KT\A and a standard set P is disjointfrom D then supp(P ) ∩A 6= ∅.

Indeed, if supp(P ) = t1, . . . , tk ⊂ T \ A and P = [t1, . . . , tk,M1, . . . ,Mk]then it follows from πT\A(D) = KT\A that there exists a point x ∈ D suchthat x(ti) ∈Mi for all i ≤ k. Therefore x ∈ D ∩ P which is a contradiction.

The set F being κ-large, there exists A1 ⊂ T with |A1| ≤ κ such that Fcovers the face KT\A1

. The property (4) shows that supp(P )∩A1 6= ∅ for anyP ∈ S0. There exists a point t1 ∈ A1 such that the family S′

0 = P ∈ S0 : t1 ∈P has cardinality κ+. Since |Mt1 | ≤ κ, we can find a family S1 ⊂ S′

0 andM1 ∈ Mt1 such that |S1| = κ+ and [t1,M1] P for any P ∈ S1.

Proceeding by induction assume that k < n and we have a set Ak ⊂ T with|Ak| ≤ κ and a family Sk such that |Sk| = κ+ and, for some t1, . . . , tk ∈ Ak andMi ∈ Mti

(i = 1, . . . , k), we have [t1, . . . , tk,M1, . . . ,Mk] P for every P ∈Sk. Therefore P = [t1, . . . , tk, s1, . . . , sn−k,M1, . . . ,Mk, E1, . . . , En−k] for everyP ∈ Sk; let Q(P ) be the set in which s1 and E1 are omitted from the definitionof P , i.e., Q(P ) = [t1, . . . , tk, s2, . . . , sn−k,M1, . . . ,Mk, E2, . . . , En−k]. It isclear that Q(P ) ≺ P ; since P is minimal, the set Q(P ) intersects F for eachP ∈ Sk.

Fix a set R ∈ Sk and let F ′ = F ∩ Q(R). The set F being κ-large, we canfind A ⊂ T with |A| ≤ κ such that F ′ covers the face KT\A and hence the faceKT\(A∪Ak) as well. Let Ak+1 = A∪Ak and observe that every set P ∈ Sk is dis-joint from F ′; this, together with (4) shows that supp(P )∩Ak+1 6= ∅. Supposefor a moment that P = [t1, . . . , tk, s1, . . . , sn−k,M1, . . . ,Mk, E1, . . . , En−k] ∈Sk and s1, . . . , sn−k ∩Ak+1 = ∅. Since F ′ covers the face KT\Ak+1

, we canfind a point x ∈ F ′ such that x(si) ∈ Ei for all i ≤ n− k; since also x(ti) ∈Mi

for all i ≤ k because x ∈ Q(R), we conclude that x ∈ F ′∩P . This contradictionimplies that s1, . . . , sn−k∩Ak+1 6= ∅ and hence the set supp(P )\t1, . . . , tkintersects the set Ak+1 for any P ∈ Sk \ R.

Therefore we can choose a family Sk+1 ⊂ Sk of cardinality κ+ together witha point tk+1 ∈ Ak+1 \ t1, . . . , tk and a set Mk+1 ∈ Mtk+1

such that we have[t1, . . . , tk+1,M1, . . . ,Mk+1] P for any P ∈ Sk+1. As a consequence, ourinductive procedure can be continued to construct a family Sn ⊂ S such that


|Sn| = κ+ while [t1, . . . , tn,M1, . . . ,Mn] P for any P ∈ Sn. Recalling thatr(P ) = n, we conclude that we have the equality P = [t1, . . . , tn,M1, . . . ,Mn]for each P ∈ Sn; this contradiction shows that |S| ≤ κ, i.e., (3) is proved.

It is straightforward that G = K \ (⋃S) is a Gκ-subset of K such that

F ⊂ G and F ∩ C = G ∩ C.

The following result generalizes Theorem 1 of [11].

Theorem 3.2. Given an infinite cardinal κ suppose that nw(Nt) ≤ κ for anyt ∈ T and C ⊂ N =

∏t∈T Nt is a dense subspace of N . Assume additionally

that we have a continuous (not necessarily surjective) map ϕ : C → L of C intoa compact space L. If y ∈ C′ = ϕ(C) and hπχ(y, L) ≤ κ then ψ(y, C′) ≤ κ.

Proof. There is no loss of generality to assume that C′ is dense in L. Choosea compact extension Kt of the space Nt for any t ∈ T ; then K =

∏t∈T Kt is a

compact extension of both N and C. There exist continuous maps Φ : βC → L

and ξ : βC → K such that Φ|C = ϕ and ξ(x) = x for any x ∈ C. It is clearthat both Φ and ξ are surjective.

For every t ∈ T fix a network Nt in the space Nt such that |Nt| ≤ κ andlet Mt = clKt

(N) : N ∈ Nt. If M = Mt : t ∈ T then the M-standardsubsets of K will be called standard. Our first step is to prove that

(5) the set Fy = ξ(Φ−1(y)) is κ-large.

Fix a point x ∈ Fy , a finite A ⊂ T and consider the set P = 〈x,A〉 =x′ ∈ K : x′(t) = x(t) for all t ∈ A. It follows from P ∩ Fy 6= ∅ thatξ−1(P ) ∩ Φ−1(y) 6= ∅ and hence y ∈ Q = Φ(ξ−1(P )). The set Q is compactand it follows from hπχ(y, L) ≤ κ that we can choose a π-base B of the spaceQ at the point y such that |B| ≤ κ. For every B ∈ B pick a set OB ∈ τ(L) suchthat ∅ 6= OB ∩Q ⊂ OB ∩Q ⊂ B. It follows in a standard way from c(K) ≤ κ

that

(6) for any U ∈ τ∗(L), the set clK(ϕ−1(U)) depends on at most κ-many coor-dinates and coincides with the set ξ(clβC(Φ−1(U))).

Apply (6) to find a set S ⊂ T of cardinality at most κ for which A ⊂ S

and the set DB = ξ(clβC(Φ−1(OB))) depends on S for any B ∈ B. The faceKT\S is residual; to show that P ∩ Fy covers KT\S fix any point w ∈ KT\S

and consider the set E = z ∈ K : πT\S(z) = w and πS(z) ∈ πS(P ). Clearly,E is a non-empty compact subset of P .

Fix any B ∈ B; it follows from OB ∩Q 6= ∅ that there is a point u ∈ ξ−1(P )such that Φ(u) ∈ OB; thus u ∈ Φ−1(OB) which shows that ξ(u) ∈ DB ∩ P .Define a point u′ ∈ K by the equalities πT\S(u′) = w and πS(u′) = πS(ξ(u)).Since the sets DB and P depend on S, we conclude that u′ ∈ DB ∩ P . On theother hand, πS(u′) ∈ πS(P ) so u′ ∈ E, and therefore E ∩DB 6= ∅.

As a consequence, Φ(ξ−1(E)) ∩OB 6= ∅ and hence Φ(ξ−1(E)) ∩B 6= ∅ forany B ∈ B; since Φ(ξ−1(E)) is a closed subset of Q and B is a π-base of Qat y, we must have y ∈ Φ(ξ−1(E)) which implies that ξ−1(E) ∩ Φ−1(y) 6= ∅

and hence E ∩ Fy 6= ∅. If v ∈ E ∩ Fy then w = πT\S(v) ∈ πT\S(P ∩ Fy); the

44 V. V. Tkachuk

point w ∈ KT\S was chosen arbitrarily so P ∩Fy covers KT\S and hence (5) isproved.

By Lemma 3.1 there exists a Gκ-set G in the space K such that Fy ⊂ G andG ∩ C = Fy ∩ C = ϕ−1(y). Therefore we can choose a family F of compactsubsets of K such that |F| ≤ κ and C \ Fy ⊂

⋃F ⊂ K \ Fy . For any F ∈ F

the set WF = L \ Φ(ξ−1(F )) is an open neighbourhood of y in L and it isstraightforward that H =

⋂WF : F ∈ F is a Gκ-subset of L such that

H ∩ C′ = y.

Corollary 3.3. Suppose that C is a dense subspace of a product N =∏

t∈T Nt

such that nw(Nt) ≤ κ for each t ∈ T . Assume that K is a compact space witht(K) ≤ κ and ϕ : C → K is a continuous (not necessarily surjective) map; letC′ = ϕ(C). Then every closed subspace of C′ is a Gκ-set, i.e., Ψ(C′) ≤ κ; inparticular, ψ(C′) ≤ κ.

Proof. Fix a non-empty closed set F ′ in the space C′ and let F = clK(F ′).Consider the quotient map p : K → KF obtained by contracting the set Fto a point and let q = p|C′. It is easy to see that we have the inequalitiest(KF ) ≤ t(K) ≤ κ; denote by y the point of the space KF represented by F

and let C′′ = p(C′). It follows from [1, Theorem 2.2.20] that hπχ(y,KF ) ≤ κ

so Theorem 3.2, applied to the map p ϕ, implies that ψ(y, C′′) ≤ κ. SinceF ′ = q−1(y), we conclude that F ′ is a Gκ-subset of C′.

Corollary 3.4. Suppose that C is a dense subspace of a product N =∏

t∈T Nt

such that nw(Nt) ≤ κ for each t ∈ T . Assume additionally that l(C) ≤ κ andK is a compact space with t(K) ≤ κ such that there exists a continuous (notnecessarily surjective) map ϕ : C → K. If C′ = ϕ(C) then hl(C′) ≤ κ.

Proof. We have l(C′) ≤ κ while every closed subspace of the space C′ is aGκ-set by Corollary 3.3. Now, a standard proof shows that hl(C′) ≤ κ.

Corollary 3.5. If C is a dense subspace of a product of cosmic spaces andK is a compact space then, for any continuous map ϕ : C → K, we haveΨ(ϕ(C)) ≤ t(K).

The last corollary has several applications in Cp-theory. Let us start withthe following observation.

Proposition 3.6. (Folklore). If the space of a topological group G embeds ina compact space of countable tightness then G is metrizable. In particular, ifCp(X) embeds in a compact space of countable tightness then Cp(X) is secondcountable and hence X is countable.

Proof. Assume that G is a dense subspace of a compact spaceK with t(K) ≤ ω.Then πχ(g,G) = πχ(g,K) ≤ ω (see [1, Theorem 2.2.20]) and hence we havethe equality χ(g,G) = πχ(g,G) = ω for any g ∈ G (see [2, Proposition 1.1]) soG is metrizable.

The following result is a curious generalization of Proposition 3.6 for thecase of condensations.


Corollary 3.7. For any X, the space Cp(X) condenses onto a space embed-dable in a compact space of countable tightness if and only if Cp(X) condensesonto a second countable space.

Proof. Apply Corollary 3.5 and the equality ψ(Cp(X)) = iw(Cp(X)).

However, it would be interesting to find out whether any continuous image ofCp(X) embeddable in a compact space of countable tightness has to be cosmicor even metrizable. It follows from Corollary 3.3 that such an image is a perfectspace. The following theorem shows that this conjecture is true when Cp(X)is a Lindelof Σ-space.

Theorem 3.8. Suppose that ϕ : Cp(X) → K is a continuous (not necessarilysurjective) map and K is a compact space with t(K) ≤ ω; let Y = ϕ(Cp(X)).Then

(i) Y is a perfect space of countable π-weight;(ii) if Cp(X) is a Lindelof Σ-space then Y is cosmic.

Proof. That Y is perfect is an immediate consequence of Corollary 3.3. Sinceω1 is a precaliber of Cp(X), it has to be also a precaliber of Y and hence

of Y . The space Y being compact, the cardinal ω1 is a caliber Y ; it followsfrom t(Y ) ≤ ω that Y has a point-countable π-base [10]. This implies thatπw(Y ) = ω and hence πw(Y ) = ω as well, i.e., we settled (i).

If Cp(X) is a Lindelof Σ-space then Cp(X) × Cp(X) is Lindelof. The spaceY × Y is a continuous image of Cp(X)×Cp(X) ≃ Cp(X ⊕X) so we can applyCorollary 3.4 to convince ourselves that Y ×Y is hereditarily Lindelof and henceY condenses onto a second countable space. This, together with the LindelofΣ-property of Y implies that nw(Y ) ≤ ω and hence (ii) is proved.

In the sequel we will need the following lemma from [13].

Lemma 3.9. If Cp(X) =⋃

n∈ω Fn and every Fn is closed in Cp(X) then thereexists n ∈ ω such that Cp(X) embeds in Fn.

Theorem 3.10. Suppose that l(Xn) = ω for all n ∈ N and Cp(X) is Lindelof.If Cp(X) condenses onto a σ-compact space Y then the space X is separableand ψ(Y ) = ω.

Proof. Fix a condensation ϕ : Cp(X) → Y and a family Kn : n ∈ ω ofcompact subsets of Y such that Y =

⋃n∈ω Kn. The set Fn = ϕ−1(Kn) is

closed in Cp(X) for every n ∈ ω. If n ∈ ω and S is an uncountable freesequence in Kn then S′ = ϕ−1(S) is an uncountable free sequence in Fn whichis impossible because l(Fn) ≤ l(Cp(X)) = ω and t(Fn) ≤ t(Cp(X)) = ω. Thiscontradiction shows that Kn has no uncountable free sequences and thereforet(Fn) ≤ ω for any n ∈ ω.

Apply Lemma 3.9 to see that there exists n ∈ ω such that C ≃ Cp(X) forsome C ⊂ Fn. Since ϕ|C maps C into Kn, Corollary 3.5 shows that ψ(ϕ(C)) ≤ω. Since ϕ|C is a condensation, we have ψ(Cp(X)) = ψ(C) ≤ ψ(ϕ(C)) = ω

and hence d(X) = ψ(Cp(X)) = ω, i.e., X is separable as promised.

46 V. V. Tkachuk

It follows from ψ(Cp(X)) = ω that Cp(X) \ f is an Fσ-set for any f ∈Cp(X). The space Cp(X) being Lindelof, Cp(X) \ f is Lindelof as well.Therefore Y \ y is Lindelof for any y ∈ Y ; this implies that ψ(Y ) ≤ ω.

Corollary 3.11. Suppose that X is an ω-monolithic space such that Cp(X)is Lindelof and Xn is Lindelof for any n ∈ N. If Cp(X) condenses onto aσ-compact space Y then nw(X) = nw(Y ) = ω.

Proof. Theorem 3.10 shows that the space X must be separable so nw(X) = ω

by ω-monolithity of X . Therefore nw(Y ) ≤ nw(Cp(X)) = nw(X) = ω.

The following result gives a complete answer (in a much stronger form) toProblems 29 and 30 from the paper [4].

Corollary 3.12. If X is an ω-monolithic compact space such that Cp(X) isLindelof and can be condensed onto a σ-compact space then X is metrizable.In particular, if X is a non-metrizable Corson compact space then Cp(X) doesnot condense onto a σ-compact space.

Corollary 3.13. Under MA+¬CH if K is a compact space such that Cp(K)is Lindelof and can be condensed onto a σ-compact space then X is metrizable.

Proof. It is a result of Reznichenko (see [3, Theorem IV.8.7]) that MA+¬CHtogether with the Lindelof property Cp(K) implies that K is ω-monolithic sowe can apply Corollary 3.12 to see that K is metrizable.

Corollary 3.14. Assume that Cp(X) is a Lindelof Σ-space and there exists acondensation of Cp(X) onto a σ-compact space Y . Then nw(X) = nw(Y ) = ω.

Proof. Denote by υX the Hewitt realcompactification of X . It is evident thatCp(X) is a continuous image of the space Cp(υX). Besides, Z = υX is aLindelof Σ-space by [8, Corollary 3.6] ; since Cp(Z) is also a Lindelof Σ-space(see [14, Theorem 2.3]), Corollary 3.11 is applicable to Z and we can concludethat nw(Z) = nw(Y ) = ω. Since X ⊂ Z, we have nw(X) ≤ nw(Z) = ω.

4. Open problems.

There are still many opportunities for discovering interesting facts aboutcondensations of function spaces. The list below shows some possible lines ofresearch in this direction.

Problem 4.1. Suppose that K is a compact space of countable tightness andϕ : Cp(X) → K is a continuous map. Is it true that ϕ(Cp(X)) is cosmic oreven metrizable?

Problem 4.2. Suppose that Cp(X) is Lindelof, K is a compact space of count-able tightness and ϕ : Cp(X) → K is a continuous map. Is it true thatϕ(Cp(X)) is cosmic or even metrizable?


Problem 4.3. Suppose that Cp(X) is hereditarily Lindelof, K is a compactspace of countable tightness and ϕ : Cp(X) → K is a continuous map. Is ittrue that ϕ(Cp(X)) is cosmic or even metrizable?

Problem 4.4. Suppose that X is compact, K is a compact space of countabletightness and ϕ : Cp(X) → K is a continuous map. Is it true that ϕ(Cp(X))is cosmic or even metrizable?

Problem 4.5. Is it true that, for any cardinal κ and any compact space K

with t(K) ≤ ω, if ϕ : Rκ → K is a continuous map then ϕ(Rκ) is cosmic or

even metrizable?

Problem 4.6. Suppose that K is a compact space of countable tightness, G isa topological group with the Lindelof Σ-property and ϕ : G→ K is a continuousmap. Is it true that ϕ(G) is cosmic?

Problem 4.7. Suppose that Cp(X) is Lindelof and there exists a condensationof Cp(X) onto a σ-compact space Y . Must Y be cosmic?

Problem 4.8. Suppose that Cp(X) is Lindelof and there exists a condensationof Cp(X) onto a σ-compact space Y . Must X be separable?

Problem 4.9. Suppose that Cp(X) condenses onto a space of countable π-weight. Must X be separable?

Problem 4.10. Suppose that Cp(X) is Lindelof and ϕ : Cp(X) → Y is acontinuous onto map. Is it true that every compact subspace of Y has countabletightness?

Problem 4.11. Suppose that K is Eberlein compact and ϕ : Cp(K) → Y isa continuous surjective map of Cp(X) onto a σ-compact space Y . Must Y becosmic?

Problem 4.12. Suppose that K is Corson compact and ϕ : Cp(K) → Y isa continuous surjective map of Cp(X) onto a σ-compact space Y . Must Y becosmic?

Problem 4.13. Suppose that X is a space such that Cp(X) has the LindelofΣ-property and ϕ : Cp(K) → Y is a continuous surjective map of Cp(X) ontoa σ-compact space Y . Must Y be cosmic?

Problem 4.14. Suppose that Cp(X) is Lindelof and Xn is also Lindelof forany n ∈ N. Assume additionally that there exists a condensation of Cp(X)onto a σ-compact space Y . Must Y be cosmic?

Problem 4.15. Suppose that K is a perfectly normal compact space. Is it truethat every σ-compact continuous image of Cp(X) has a countable network?

48 V. V. Tkachuk

References

[1] A. V. Arhangel’skii, The structure and classification of topological spaces and cardinalinvariants (in Russian), Uspehi Mat. Nauk 33, no. 6 (1978), 29–84.

[2] A. V. Arhangel’skii, On relationship between invariants of topological groups and theirsubspaces (in Russian), Uspehi Mat. Nauk 35, no. 3 (1980), 3–22.

[3] A. V. Arhangel’skii, Topological Function Spaces, Kluwer Academic Publishers, Mathe-matics and Its Applications 78, Dordrecht 1992.

[4] A. V. Arhangel’skii and O.I. Pavlov, A note on condensations of Cp(X) onto compacta,(in Russian), Comment. Math. Univ. Carolinae 43, no. 3 (2002), 485–492.

[5] F. Casarrubias–Segura, On compact weaker topologies in function spaces, TopologyAppl. 115 (2001), 291–298.

[6] R. Engelking, General Topology, PWN, Warszawa, 1977.[7] W. Marciszewski, A function space Cp(X) without a condensation onto a σ-compact

space, Proc. Amer. Math. Soc. 131, no. 6 (2003), 1965–1969.[8] O. G. Okunev, On Lindelof Σ-spaces of continuous functions in the pointwise topology,

Topology Appl. 49, no. 2 (1993), 149-166.[9] E. G. Pytkeev, Upper bounds of topologies, Math. Notes 20, no. 4 (1976), 831–837.

[10] B. E. Shapirovsky, Cardinal invariants in compact spaces (in Russian), Seminar Gen.Topol., ed. by P.S. Alexandroff, Moscow Univ. P.H., Moscow, 1981, 162–187.

[11] L. V. Shirokov, On bicompacta which are continuous images of dense subspaces of topo-logical products (in Russian), Soviet Institute of Science and Technology Information(VINITI), Registration number 2946–81, Moscow, 1981.

[12] M. G. Tkachenko, Factorization theorems for topological groups and their applications,Topology Appl. 38 (1991), 21–37.

[13] V. V. Tkachuk, Decomposition of Cp(X) into a countable union of subspaces with ”good”properties implies ”good” properties of Cp(X), Trans. Moscow Math. Soc. 55 (1994),239–248.

[14] V. V. Tkachuk, Behaviour of the Lindelof Σ-property in iterated function spaces, Topol-ogy Appl. 107 (2000), 297–305.

Received January 2008


Vladimir V. Tkachuk ([email protected])Departamento de Matematicas, Universidad Autonoma Metropolitana, Av.San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa, C.P. 09340, Mexico D.F.



Volume 10, No. 1, 2009

pp. 49-68

Pointwise convergence and Ascoli theorems fornearness spaces

Zhanbo Yang∗

Abstract. We first study subspaces and product spaces in the

context of nearness spaces and prove that U-N spaces, C-N spaces, P-

N spaces and totally bounded nearness spaces are nearness hereditary;

T-N spaces and compact nearness spaces are N -closed hereditary. We

prove that N2 plus compact implies N -closed subsets. We prove that

totally bounded, compact and N2 are productive. We generalize the

concepts of neighborhood systems into the nearness spaces and prove

that the nearness neighborhood systems are consistent with existing

concepts of neighborhood systems in topological spaces, uniform spaces

and proximity spaces respectively when considered in the respective

sub-categories. We prove that a net of functions is convergent under the

pointwise convergent nearness structure if and only if its cross-section

at each point is convergent. We have also proved two Ascoli-Arzela

type of theorems.

2000 AMS Classification: 54E17, 54E05, 54E15, 68U10

Keywords: Nearness spaces, subspace, product space, neighborhood system,pointwise convergent, Ascoli’s theorem

1. Introduction

As a natural extension of geometry, the concept of “near/apart” has been acenter for topology and related studies. Topology characteries the “nearness”between a point and a set. Proximity [14] is an axiomatization of “nearness”between two sets. Contiguity [10] describes the concept of nearness among theelements of a finite family of sets. The concept of “nearness space” introducedby Herrlich [8] in 1974 attempts to characterize the nearness of an arbitrarycollection of sets. The category of nearness spaces, the most general amongthe aforementioned structures, can be used as a unifying framework. The

∗This work was in part supported by a grant from the 2008 Faculty Research Fund of theUniversity of the Incarnate Word.

50 Z. Yang

categories of several aforementioned structures can all be “nicely embedded”into the category of the nearness spaces as (either bireflective or bicoreflective)sub-categories ([8]).

In recent years, the notion of “nearness” in a number of different variationshas found new applications in digital topology, image processing and patternrecognition areas, perhaps due to the fact that those structures are “richer”than classical topology. In 1995, Latecki and Prokop [12] used a weaker ver-sion of proximity spaces called semi-proximity spaces (sp-spaces). They talkedabout the possibility of describing all digital pictures used in computer visionand computer graphs as non-trivial semi-proximity spaces, which is not possiblein classical topology. They also discussed the application of “semi-proximitycontinuous functions” in well-behaved operations such as thinning on digitalimages. In 1996, Chaudhuri [4] introduced a new definition for the neighborsof an arbitrary point P . This new definition used a “centroid criterion” tocapture the idea that the neighbors of P should be as near to P and as sym-metrically paced around P as possible. This new definition could be used forpattern classification, clustering and low-level description of dot patterns. In2000, Ptak and Kropatsch [16] discussed the application of proximity spacesin studying of digital images. They showed by examples that the “proximatecomplexity” of a finite covering in a digital picture might be too high to beadequately depicted in a finite topological space, which might indicate anotherconceptual advantage of proximities over topologies. Most recently, in 2007,Wolski and Peters [18, 15] investigated approximation spaces in the contextof topological structures which axiomatized certain notion of nearness. Peters[15] pointed out particularly that the concept of “nearness” was not confined tospatial nearness, or geometrical likeness. It was possible to introduce a nearnessrelation that could be used to determine the “nearness” among sets of objectsthat were spatially far away and, yet, “qualitatively” near to each other.

The main objectives of this paper are to establish a ”pointwise convergent”nearness structure on a function space made of a family of functions from Xto Y and to establish two versions of the Ascoli-Arzela theorems for nearnessspaces that relate the compactness of the underlining space Y with that of thefunction space. Since the function space is really a subspace of the productspace Y X , we begin with the nearness structures on a subspace and discuss thehereditary properties for a number of important concepts in nearness spaces.We then define the nearness structure on a product spaces and discuss its var-ious properties. The nearness structure on a function space is then introducedas a subspace of the product space Y X . We will end the discussion with twoAscoli-Arzela type of theorems.

Some work in the past, such as [2] (1979), [7] (1979) and [1] (2006) havediscussed a number of results with respect to subspaces and product spaces.Most of the results in that paper were dealing with topological nearness (T-N )spaces and do not duplicate what are to be presented in this paper. For thepurpose of clarity and being self-contained, we will still give the definition of”subspace” and ”product” space here and prove relevant results.

Pointwise convergence for nearness spaces 51

The classical Ascoli-Arzela theorem was proved in the 19th century firstby Ascoli and then independently by Arzela. It characterizes compactness ofsets of continuous real-valued functions on the interval [0,1] with respect tothe topology of uniform convergence. It is commonly known that the issuecame from the fact that a convergent sequence of continuous functions maynot converge to a continuous function. So the natural question is: under whatconditions the limit of a convergent sequence of continuous functions is stillcontinuous. It turned out that the concept of equicontinuous was used tocharacterize the condition needed (see [11]) in topological spaces. In 1970,[13] discussed Ascoli’s theorem for the spaces of multifunctions. In 1981, [6]discussed Ascoli’s theorem for topological categories. In 1984, [3] discussedAscoli’s theorem for a class of merotopic spaces. In 1993, [5] studied a versionof the Ascoli’s theorem for set valued proximally continuous functions. In 2001,[17] proved a version of Ascoli’s theorem for sequential spaces. As far as weknow, no nearness space version of the Ascoli’s theorem has been establishedyet at this time.

We have practical reason to be interested in this topic. In many cases, adigital image processing algorithm is essentially the application of a sequenceof deformation functions to a digital plane. For example, [12] proved that adeletion of a simple point (a point that does not affect the connectness of thedigital picture) can be regarded as a sp-continuous function. Hence a thinningalgorithm that preserves connectness can be arranged as a sequence of sp-continuous functions. We may be able to use the tools of function spaces, andthe results on convergence of function sequences to study the image processingalgorithms, which opens a new set of doors.

The rest of this paper is organized as follows: Section 2 is a collection ofthe major definitions involving nearness spaces that are relevant to this paper.Section 3 studies the nearness structures on a subspaces. Section 4 is aboutthe product spaces and the function spaces. The summary at the end concludesthis paper.

2. Notation and Definitions

In this section, we define the basic concepts used throughout this paper.We will use the language of Categories in some of our discussions. For

readers who are not familiar with Category theory, a category is basically afamily of objects with a particular type of structures. For example, we cantalk about the category of all topological spaces, the category of all groups,etc. The so called “morphism” from one object to another is a function thatpreserves the structure on the objects. For example, a “morphism” in thecategory of topological spaces would be a continuous function. A ”morphism”in the category of all groups would be a homomorphism. An embedding fromone category into another category is a way to assign each object from onecategory to an object of the other category in some injective manner that alsopreserves the morphisms. For readers who are interested at further informationabout category theory, please see [20].

52 Z. Yang

The readers can see Kelley [11] or any common general topology text bookfor terms in general topology.

2.1. Basic Notations. Let X be a set, P(X) represents the power set of X .P0(X) = X,P1(X) = P(X), . . . ,Pn(X) = P(Pn−1(X))

A, B, . . . represent elements in P(X), i.e. subsets in XA , B, . . . represent elements in P2(X), i.e. subsets in P(X)ξ, η, . . . represent elements in P3(X), i.e. subsets in P2(X)A C = X − A : A ∈ A For each B ⊆ X , A (B) =

⋃

A : A ∩ B 6= φ, A ∈ A ξA denotes A ∈ ξ, ξA denotes A /∈ ξAξ B denotes A, B ∈ ξ. Aξ B denotes A, B /∈ ξclξA = x : x ξ A, intξA = X − clξ(X − A)clξA = clξA : A ∈ A , intξA = intξA : A ∈ A A ∨ B = A ∪ B : A ∈ A , B ∈ BA ∧ B = A ∩ B : A ∈ A , B ∈ BA ≺ B if and only if for any B ∈ B , there is A ∈ A such that A ⊆ B.

This is referred to as ”B co-refines A ”.

2.2. Definitions Related to Nearness Structure. We will restate somemajor definitions about nearness spaces here (due to [8]):

(i) Let X be a non-empty set. The ordered pair (X, ξ) is said to be anearness space, or N -space, if the following are satisfied:

(N1) If⋂

A 6= φ, then ξA .(N2) If ξB, and for each A ∈ A , there exists a B ∈ B such that

B ⊆ clξA, then ξA , i.e. B ≺ clξA .

(N3) If ξA and ξB, then ξ(A ∨ B).(N4) If φ ∈ A, then ξA .

(ii) Let X be a set. Let (X, ξ) and (Y, η) be two N-spaces. A functionf : X → Y is said to be a (ξ, η) N-preserving map, or an N-preservingmap, or simply an N-mapping, if one of the following two equivalentconditions is satisfied:

(M1) If ξA , then ηf(A ), where f(A ) = f(A) : A ∈ A

(M2) If ηB, then ξ f−1(B), where f−1(B) = f−1(B) : B ∈ B.We will use the notation NEAR to represent the category of all

nearness spaces with N-mappings.(iii) An N-space is called an N1-space, if the following is satisfied:

(N0) If xξy, then x = y.(iv) An N-space is called an N2 space, if for any x, y ∈ X , x 6= y implies the

existence of A ⊆ X and B ⊆ X such that A ∩ B = φ, ξx, X − Aand ξy, X − B.

(v) An N-space is called a T-N space, if the following is satisfied:(T) If ξA , then

⋂

clξA 6= φ.A T −N2 space is an N2 space that also satisfies the condition (T).


We will use the symbol T − NEAR to represent the subcategory ofNEAR, consists of all T-N spaces with N-mappings.

(vi) An N-space is called a U-N space, if the following is satisfied:

(U) If ξA , then there exists a B such that ξB and for each B ∈ B,there is an A ∈ A such that BC(X − B) ⊆ X − A.

We will use the notation U − NEAR to represent the subcategory ofNEAR, consists of all U-N spaces with N-mappings.

(vii) An N-space is called a C-N space, if the following is satisfied:(C) If ξA , then there is a finite subcollection B ⊆ A such that ξB.We will use the notation C − NEAR to represent the subcategory of

NEAR, consists of all C-N spaces with N-mappings.(viii) An N-space is called a P-N space, if it satisfies both of the conditions

(U) and (C).We will use the notation P − NEAR to represent the subcategory of

NEAR, consists of all P-N spaces with N-mappings.(ix) An N-space is called a totally bounded space, if one of the following

equivalent conditions is satisfied:(B1) If ξA , then there is a finite subcollection B ⊆ A such that

⋂

B = φ.(B2) If F is a filter on X , then ξF .

(x) An N-space is called a compact space, if it satisfies both condition (T)and (C).

(xi) Let X be a set. ξα : α ∈ Λ is a family of N-structures on X . The leastupper bound, denoted by ξ = supξα : α ∈ Λ, is defined as follows:

ξA if and only if there are finitely many Ai’s such that for each

i = 1, 2, ...n, ξiAi, and A ≺n∨

i=1Ai.

(xii) Let (X, ξ) be a nearness space. Then the topology induced by theclosure operator A 7→ clηA is denoted by Tξ.

3. Nearness Structure on Subspaces

We will begin by giving the definition of a ”nearness subspace”, then proceedto show that subspaces as defined here are well-defined (Theorem 3.2), act”natural” (Lemma 3.3) and produces a topology that is consistent with thesubspace topology (Theorem 3.8).

Definition 3.1. Let (X, ξ) be a nearness space and X0 ⊆ X. Define ξ0 on X0

as follows:

ξ0 = A ⊆ P(X0) : ξA .

We will denote such ξ0 as ξ0 = ξ|X0and refer to it as the “nearness structure

on the subspace induced by the nearness structure ξ”.

Theorem 3.2. Let (X, ξ) be a nearness space, and X0 ⊆ X. ξ0, as defined inDefinition 3.1, is a nearness structure on X0.

54 Z. Yang

The proof is an easy deduction from the fact that ξ0 is consists of the typeof A that are in ξ already. We will skip the details. The following lemma isalso easy to prove.

Lemma 3.3. Let (X, ξ) be a nearness space and X0 ⊆ X. Let i : X0 → X bethe inclusion map, then ξ0 = i−1(ξ). Hence i : X0 → X is N-preserving.

Lemma 3.4. Let X be a set, (Y, η) be a nearness space. Let f : X → Y be aninjective map. Then

Tf−1(η) = f−1(Tη).

Proof.

clf−1(η)A = x : xf−1(η)A

= x : f(x)η f(A)

= x : f(x) ∈ clη(f(A))

= x : x ∈ f−1(clη(f(A))

= f−1(clη(f(A)).

We will next exam whether some of the common properties are hereditary.i.e. whether a particular condition or property can be “inherited” by its sub-spaces from their “mother” spaces. It turns out that being a T-N space is nothereditary (Example 3.5). Neither was being a compact N-space. Those twoproperties can be inhered by N-closed subspaces. Many of the other propertiesare hereditary.

Example 3.5. A subspace of a T-N space may not be a T-N space.

Let X = R, the real line with an ordinary open interval topology T . Then(X, T ) is a R0- space, hence corresponding to a T-N space (X, ξ) ([8] Theorem2.2). Now we let X0 = X − 0, A = (−∞, 0) and B = (0,∞). Then clξA ∩clξB 6= φ, but clξ0

A ∩ clξ0B = φ. This means that the nearness subspace

(X0, ξ0) does not satisfy condition (T), hence not an T-N space.

Definition 3.6. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace.(X0, ξ0) is said to be a N-closed subspace, if for any A ⊆ X0, we have clξ0

A =clξA.

Theorem 3.7. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace. Then(X0, ξ0) is a N-closed subspace if and only if clξ0

X0 = clξX0.

Proof. The necessity is obvious. We now will prove the sufficiency. Take anyA ⊆ X0, then

clξ0A = clξA ∩ X0

= clξA ∩ clξX0

= clξ(A ∩ X0)

= clξA.


Theorem 3.8. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace. Then(a) Tξ0

= Tξ|X0

.

(b) If (X, ξ) is a T-N space and (X0, ξ0) is an N-closed subspace, then (X0, ξ0)is also a T-N space.

Proof. (a) Tξ0= Ti−1(ξ) = i−1(Tξ) = Tξ

∣

∣

X0

.

(b) Let ξ0A0, then ξA0. It follows from the assumption of (X0, ξ0) being anN-closed subspace that clξA0 = clξ0

A0. Therefore⋂

clξ0A0 : A0 ∈ A0 =

⋂

clξA0 : A0 ∈ A0 6= φ. So (X, ξ0) satisfies the condition (T).

Theorem 3.9. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace.Then if (X, ξ) is a U-N space, so is (X0, ξ0). Moreover, Uξ0

= Uξ|X0

, whereUξ0

and Uξ denotes the uniformity induced by the nearness structure ξ0 and ξrespectively. Uξ|X0

is the uniformity Uξ restricted to X0.

Proof. If ξ0A0, then ξA0. Since (X, ξ) is an U-N space, there exists a ξB thatsatisfies the condition (U) with respect to A0. Let B0 = B ∩ X0 : B ∈ B.

From (N2) we can see that ξ0B0. For each B0 ∈ B0 = B ∩ X0 : B ∈ B,there is a B ∈ B such that B = B ∩ X0. So by condition (U), there shouldbe an A0 ∈ A0 such that A0 ⊆

⋂

C : B ∪ C 6= X, C ∈ B. Also becauseA0 ⊆ X0, we have

A0 = A0 ∩ X0

⊆⋂

C : B ∪ C 6= X, C ∈ B ∩ X0

=⋂

C ∩ X0 : B ∪ C 6= X, C ∈ B

⊆⋂

C0 : B0 ∪ C0 6= X0, C0 ∈ B0

Therefore, ξ0 satisfies the condition (U). Furthermore, for any A0 ∈ P2(X), wehave the following equivalent deductions:

A0 ∈ U |i−1(ξ)

⇔ AC0 /∈ i−1(ξ)

⇔ AC0 /∈ ξ0

⇔ A0 ∈ U |ξ0

⇔ A0 ∈ i−1(Uξ).

Therefore,

Uξ0= Ui−1(ξ) = i−1(Uξ) = Uξ|X0

.

56 Z. Yang

Theorem 3.10. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace. If(X, ξ) is a C-N space, so is (X0, ξ0). Moreover, Cξ0

= Cξ|X0

, where Cξ0and Cξ

denote the contiguity induced by the nearness structure ξ0 and ξ respectively.Cξ|X0

is the contiguity Cξ restricted to X0.

Proof. For any A0 ∈ P2(X) and ξ0A0, then ξA0. By condition (C), A0 has afinite subcollection B0 such that ξB0. This implies that ξ0B0. And further-more,

A0 ∈ Cξ0

⇔ A0 ∈ ξ0 and A0 is finite.

⇔ A0 ∈ ξ, A0 is finite

⇔ A0 ∈ Cξ

⇔ A0 ∈ Cξ|X0

.

Since a P-N space is one that satisfies condition (U) and (C), the followingTheorem is obvious from the Theorems 3.9 and 3.10:

Theorem 3.11. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace.Then if (X, ξ) is a P-N space, so is (X0, ξ0).

Since a compact nearness space is one that satisfies condition (T) and (C),the following theorem is obvious from Theorems 3.8 and 3.10:

Theorem 3.12. Let (X, ξ) be a nearness space and (X0, ξ0) be a N-closedsubspace. Then if (X, ξ) is a compact nearness space, so is (X0, ξ0).

Lemma 3.13. Let (X, ξ) be a T-N space. Then (X, clξ) is topologically compactif and only if (X, ξ) is a compact nearness space.

Proof. Recall that (X, ξ) is a compact nearness space if and only if condition(T) and (C) are satisfied.

Necessity: If (X, clξ) is topologically compact. Take an A such that ξA .We want to show that condition (C) is met by showing that A has a finite

subcollection B such that ξB. We will first claim that⋂

clξA = φ. If not,then by (N1), ξclξA would be true. For each A ∈ A , there would be aclξA ∈ clξA such that clξA ⊆ clξA. By (N2), ξA would be true, and that

would contradict to the assumption of ξA . So⋂

clξA = φ must be true.(clξA )C is an open cover of X . Since (X, clξ) is topologically compact, wewill let B be the finite subcollection of A and (clξB)C is an open cover of X .

This implies that⋂

clξB = φ. By condition (T), ξB is true. Hence condition(C) has been met.Sufficiency: If (X, ξ) is a compact nearness space, which means that itmeets condition (T) and (C). Any open cover of (X, clξ) can be expressed asthe complement collection of a collection clξA and

⋂

clξA = φ. From

condition (T), ξclξA is true. From condition (C), there must be a finite


subcollection of clξA , say clξB, such that ξclξB is true. It follows that⋂

clξB = φ. Then (clξB)C is the finite subcover of the original open cover.Hence (X, clξ) is topologically compact.

From Lemma 3.13, one can easily see the following is true:

Lemma 3.14. Let (X, ξ) be a compact nearness space, (Y, η) be a T-N spaceand f : (X, ξ) → (Y, η) be N-preserving, then (Y, η) is a compact nearnessspace.

Definition 3.15. Let X be a set. Define a partial order among all possiblenearness structures on X as follows: If ξ1 and ξ2 are two nearness structureson a set X, then ξ1 ≤ ξ2 if and only if ξ1 ⊇ ξ2.

Lemma 3.16. (i) Let X be a set and ξ1 and ξ2 be two nearness structureson a set X. ξ2 ≤ ξ1 if and only if i : (X, ξ1) → (X, ξ2) is N-preserving.

(ii) Let (X, ξ) and (Y, η) be two nearness spaces. Then f : (X, ξ) → (Y, η)is N-preserving if and only if ξ ≥ f−1(η).

Proof. (i)

ξ2 ≤ ξ1

⇔ ξ2 ⊇ ξ1

⇔ i(ξ1) ⊆ ξ2

⇔ i : (X, ξ1) → (X, ξ2) is N − preserving.

(ii)

f : (X, ξ) → (Y, η) is N − preserving

⇔ ξ ⊆ f−1(η)

⇔ ξ ≥ f−1(η).

Theorem 3.17. Let (X, ξ) be a T − N2 space and (X0, ξ0) be a N-compactsubspace. Then (X0, ξ0) is N-closed.

Theorem 3.18. If (X, ξ) is totally bounded, and X0 ⊆ X, then (X0, ξ0) istotally bounded.

Proof. ξ0A0 implies that ξA0. Hence, by condition (B1) for (X, ξ), there isa B0, a finite subcollection of A0, such that

⋂

B0 = φ. So (X0, ξ0) satisfies(B1).

The following lemma proved by Hunsaker and Sharma as Corollary (2.5) intheir 1974 paper [9] is used to prove the next theorem.

Lemma 3.19. Let f : (X, ξα) → (Y, ηα) be an N-preserving map for eachα ∈ Λ. Then f : (X, supξα) → (Y, supηα) is an N-preserving map.

58 Z. Yang

The following theorem is needed to ensure that the concept of nearnessstructure on the function space, which will be introduced in next section, iswell defined.

Theorem 3.20. If ξα : α ∈ Λ is a family of nearness structures on X andX0 ⊆ X. Then

supα∈Λ

ξα

∣

∣

∣

∣

X0

= supα∈Λ

ξα|X0.

Proof. Let i : X0 → X be the inclusion map. For each α ∈ Λ, By Lemma 3.3,

i : (X0, ξα|X0) → (X, ξα)

is N-preserving. From Lemma 3.19,

i : (X0, supα∈Λ

ξα

∣

∣

∣

∣

X0

) → (X, supα∈Λ

ξα)

is still N-preserving. By Lemma 3.16,

supα∈Λ

ξα

∣

∣

∣

∣

X0

≥ i−1(supα∈Λ

ξα) = supα∈Λ

ξα

∣

∣

∣

∣

X0

.

Moreover, for each α ∈ Λ,

supα∈Λ

ξα ⊆ ξα,

hence

supα∈Λ

ξα

∣

∣

∣

∣

X0

⊆ ξα|X0.

It follows that

supα∈Λ

ξα

∣

∣

∣

∣

X0

⊆ supα∈Λ

ξα

∣

∣

∣

∣

X0

.

i.e.

supα∈Λ

ξα

∣

∣

∣

∣

X0

≥ supα∈Λ

ξα|X0.

Therefore,

supα∈Λ

ξα

∣

∣

∣

∣

X0

= supα∈Λ

ξα|X0.

4. The Pointwise Convergent Nearness Structure on Function

Space

The following theorem shows that the least upperbound nearness structure isa generalization of the respective least upperbound structure when consideredin each of the subcategory of T − NEAR, U − NEAR and P − NEAR respec-tively.


Figure 1. Commutative diagram of the natural projections

Theorem 4.1. If ξα : α ∈ Λ is a family of nearness structures on X and letξ = sup

α∈Λξα. Then, when considered in T − NEAR, U − NEAR, or P − NEAR,

ξ will induce the least upper bound of the respective structures induced by ξα :α ∈ Λ in the respective type of spaces.

Proof. Let F be the isomorphic functor from T − NEAR to R0 − TOP. We justneed to prove that F is order preserving. Assume ξ1 ≤ ξ2. By Lemma 3.16,ξ1 ≤ ξ2 ⇔ i : (X, ξ2) → (X, ξ1) is N-preserving ⇔ F (i) is a morphism fromF [(X, ξ2)] to F [(X, ξ1)]1) ⇔ Tξ2

⊇ Tξ1. This shows that F does preserve the

order.The proofs for other two types are parallel and therefore omitted.

The following theorem ensures that the product nearness structure in thecategorical sense is the largest nearness structure on the product space thatmakes all natural projections N-preserving.

Theorem 4.2. If (Xα, ξα) : α ∈ Λ is a family of nearness spaces and let Pα :∏

α∈Λ Xα → Xα be the natural projection map. Then the nearness structure

ξ∗ = supα∈Λ

P−1α (ξα) is exactly the categorical product of (Xα, ξα) : α ∈ Λ.

Proof. It would suffice to show that for any N-space (X, ξ), and any family ofN-preserving maps fα : X → Xα : α ∈ Λ, there is an unique N-preservingmap f : X →

∏

α∈Λ Xα such that ∀α ∈ Λ, Pα f = fα. i.e. the diagram inFigure 1 is commutative.

We will first make a claim that for any map f : X →∏

Xα, f is N-preservingif and only if for each α ∈ Λ, Pα f is N-preserving. In fact, the necessity isobvious. Let us assume that for each α ∈ Λ, Pα f is N-preserving. FromLemma 3.19, f is (ξ, sup

α∈ΛP−1

α ) N-preserving. i.e. it is (ξ, ξ∗) N-preserving.

Now we consider the family of N-preserving maps fα : X → Xα : α ∈ Λ.Define f in the natural way (usually known as the “evaluation map”): ∀x ∈X, f(x) = (fα(x))α∈Λ. Then Pα f = fα. Since each fα is N-perserving, eachPα f is N-preserving. By earlier proof, f is N-preserving. We also know thatsuch an f is unique from its definition.

60 Z. Yang

The following purely categorical lemma should be obvious:

Lemma 4.3. If C is a category, Aα : α ∈ Λ is a family of objects.∏

α∈Λ Aα

is the categorical product in C. D is another category isomorphic to C with F

as the isomorphic functor from C to D. Then F (∏

α∈Λ Aα) =∏

α∈Λ F (Aα).

We now officially define the nearness structure on the function space:

Definition 4.4. If (Y, ξ) is a N-space, Xis a non-empty set. F ⊆ Y X . Ifwe consider Y X as a product and let ξ∗ be the product nearness structure asdefined in Theorem 4.2. Let ξρ = ξ∗|

F. Then ξρ is said to be the pointwise

convergent nearness structure on F ⊆ Y X .

Notice that if ex : Y X → Y : x ∈ X is the family of natural projections,

then ξ∗ = supx∈X

e−1x (ξ). So by Theorem 3.20, ξρ = ξ∗|

F= sup

x∈X

e−1x (ξ)

∣

∣

∣

∣

F

=

supx∈X

e−1x (ξ)

∣

∣

F.

The readers may refer to [11] for the concept of product topology, productuniformity, pointwise convergent topology and pointwise convergent uniformity.Refer to [14] for the concepts of product proximity and pointwise convergentproximity.

We would like to make sure that the product nearness structure and thepointwise convergent nearness structure is a generalization of the respectivestructures in topological spaces, uniform spaces and proximity spaces respec-tively.

Theorem 4.5. When considered in each of the subcategory T − NEAR,U − NEAR, or P − NEAR,

(i) ξ∗ will induce the Tychonoff product topology, product uniformity orthe product proximity respectively.

(ii) ξρ will induce the pointwise convergent topology, the pointwise conver-gent uniformity or the pointwise convergent proximity respectively

Proof. The first conclusion can be obtained from Theorems 4.1 and 4.2. Thesecond conclusion can be obtained from Theorems 3.8, 3.9 and 3.10.

Next we will try to generalize the concept of ”neighborhood”, which is es-sential when characterizing ”convergence”.

Definition 4.6. If (X, ξ) is a N-space, and A ⊆ X. A subset U is called anearness neighborhood of A if there is a ξA such that A ⊆ A C(A) ⊆ U . Thenotation NearN(A) represents the collection of all nearness neighborhood of asubset A. If the set A contains only one point x, then we simplify the notationfrom NearN(x) to NearN(x).

If there are two types of neighborhood system on a space that characterizethe same convergence, i.e. being convergent under one neighborhood systemis equivalent to being convergent under the other neighborhood system, thenwe consider them as ”equivalent” neighborhood systems. This is typically


characterized by the condition that any neighborhood under one system alwayscontains a neighborhood in the other system. For example, consider the twodimensional Cartesian plane. The neighborhood of circular disks centered at apoint is equivalent to the neighborhood of squares centered at the same point.

We will try to show that the nearness neighborhood generalizes the topo-logical neighborhood([11]), uniform neighborhood ([11]), and proximal neigh-borhood ([14]) by showing that the nearness neighborhood system is reallyequivalent to the respective neighborhood system in the respective subcate-gories.

Theorem 4.7. Let (X, ξ) be a N-space, x ∈ X and A ⊆ X.

(i) In T − NEAR, NearN(x) is equivalent to a topological neighborhoodsystem at x.

(ii) In U − NEAR, NearN(x) is equivalent to a uniform neighborhood sys-tem at x.

(iii) In P − NEAR, NearN(A) is equivalent to a proximal neighborhood sys-tem at A.

Proof. (i) Take an U ∈ NearN(x), there will be an A /∈ ξ such thatx ∈ A C(x) ⊆ U . For this A , by conditions (N1) and (N2),

⋂

clξA =∅. So (clξA )C is a cover of X . There will be an A ∈ A such thatx0 ∈ X − clξA ⊆ X − A ⊆ A C(x0). The set X − clξA is an openneighborhood of x.

On the other hand, if we take an open neighborhood V of x underthe topology Tξ, then x /∈ X − V and X − V is closed, i.e. clξ(X −V ) = X − V . Let A = x, X − V , then A /∈ ξ. And A C(x) =X − (X −V ) = V ⊆ V . This shows that V is a nearness neighborhoodalso.

(ii) Take an U ∈ NearN(x), there will be an A /∈ ξ such that x ∈ A C(x) ⊆U . For this A , since A /∈ ξ, A C is a cover of X , which means thatthere has to be an A ∈ A such that x ∈ X − A. Let

UA = ⋃

A∈A

((X − A) × (X − A)) : A /∈ ξ,

then

UA [x] = y : (x, y) ∈ UA

= y : ∃A ∈ A ∋ (x, y) ∈ (X − A) × (X − A)

= y : ∃A ∈ A ∋ x ∈ (X − A), y ∈ (X − A)

=⋃

X − A : x ∈ X − A

= AC(x)

This “equal” relation shows the equivalency between the uniform neigh-borhood system and the nearness neighborhood system.

(iii) [14] stated that a set B is a proximal ( δ−) neighborhood of a set A ifA ≪ B. In [19], the proximity ≪ξ induced by a nearness structure is

62 Z. Yang

defined as A ≪ξ B if and only if there is a A /∈ ξ such that A C(A) ⊆ B.So the equivalency between the nearness neighborhood system and theproximal neighborhood systems is obvious.

The following theorem demonstrates the consistency of the N -converges withthose previously established concepts of convergent nets. With the establish-ment of the previous theorem, the proof should be obvious.

Theorem 4.8. The N-convergence, as it is defined in Definition 4.9, whenconsidered in T − NEAR, U − NEAR and P − NEAR, is equivalent to the con-vergence with respect to the corresponding types of structures, respectively.

A set D is said to be a directed set, if it is endowed with a reflexive andtransitive binary relation ≥ such that ∀m, n ∈ D, ∃p ∈ D s.t. p ≥ m and p ≥ n.i.e. for any two elements of D, there is always another element that precedesthem. (see [11])

As a generalization of sequences, a net in a set X is a function x : D → X ,where D is a directed set. We typically write a net as xd : d ∈ D.

We would like to exam the relation of the convergency of a net of functionsand that of the nets obtained by fixing the net of functions at any arbitrarypoint x of X . Of course, we expect the two convergences are to be equivalent.Theorem 4.7 shows exactly that.

Definition 4.9. If (X, ξ) is a N-space, xd : d ∈ D is a net in X. We sayxn : n ∈ D N-converges to a point x0 ∈ X, if for any ξA , there is an N ∈ Dsuch that for each n ≥ N, n ∈ D, we have xn ∈ A C(x0).

Theorem 4.10. If (Y, ξ) is a N-space, Xis a non-empty set. fn : n ∈ D isa net in F ⊆ Y X . Then fn : n ∈ D N-converges to a function f in (F , ξρ)if and only if for any x ∈ X, fn(x) : n ∈ D, as a net in (Y, ξ), N-convergesto the point f(x).

Proof. Necessity: Take an arbitrary point x ∈ X , take an ξA , since thenatural projection map

ex : Y X → X

is N-preserving, we have ξρe−1x (A ), so ξρ(e−1

x (A )∣

∣

F). Since fn : n ∈ D is

N-convergent to f in F . There is an n ∈ D, such that for eachm > n, m ∈ D, we have

fm ∈ (e−1x (A )

∣

∣

F)C(f).

i.e. There is an A ∈ A , such that

fm, f ⊆ F − e−1x (A).

Sofm(x) = ex(fm) /∈ A

andf(x) = ex(f) /∈ A.


i.e.

fm(x) ∈ X − A,

and

f(x) ∈ X − A

Therefore,

fm(x) ∈ AC(f(x)).

Sufficiency: Assume that fn : n ∈ D is a net in F ⊆ Y X . Furthermore,for any x ∈ X , assume that fn(x) : n ∈ D, as a net in X , N-converges tothe point f(x). We now arbitrarily take a B such that ξρB. By the definitionof ξ∗ as the least upper bound, there should be finitely manyBi ⊆ P(X), i = 1, 2, ..., n, such that for each i, we have Bi /∈ e−1

x (ξ)∣

∣

F, and

B ≺ ∨Bi. ξexi(Bi), so there is an mi such that for any n ≥ mi, there should

be an Ai ∈ exi(Bi), fn(xi), f(xi) ⊆ X − Ai. Since Ai ∈ exi

(Bi), thereexists a Bi ∈ Bi and Ai = exi

(Bi). So fn(xi) /∈ exi(Bi), and f(xi) /∈ exi

(Bi).i.e. exi

(fn) /∈ exi(Bi), and exi

(f) /∈ exi(Bi). Therefore, fn, f ⊆ F − Bi, or

we can say that fn ∈ BCi (f). Since there are only finitely many mi’s. We will

let N = maxm1, ..., mn. Then for any n ≥ N ,fn ∈ ∧(BC

i )(f) = (∨Bi)C(f). Hence fn ∈ BC(f).

The following corollary is associated with the concept of ”accumulationpoints” in classical topology.

Corollary 4.11. If (X, ξ) is a N-space, B is a subset of X, xd : d ∈ D is anet in B. Then

(i) If xd : d ∈ D is N-convergent to x0 ∈ X, then x0 ∈ clξB.(ii) In T − NEAR, for any x0 ∈ clξB, there is a xd : d ∈ D in B and

xd : d ∈ D is N-convergent to x0.

Proof. (i) If, to the contrary, x0 /∈ clξB. Then A = x0, B /∈ ξ. Sincexd : d ∈ D, and A C = X − x0, X − B, there is an N ∈ D suchthat for any n ≥ N , xn ∈ A C(x0) = X − B. But this contradicts tothe assumption that xd : d ∈ D ⊆ B. Hence x0 ∈ clξB must be true.

(ii) In T − NEAR, from Theorem 4.8(i), N-convergence is equivalent totopological convergence and clξB is the topological closure of the setB. The conclusion must be true due to classical topology.

The following Corollary is a natural consequence of the Corollary 4.11:

Corollary 4.12. Let (X, ξ) be a N-space and B ⊆ X.

(i) If B is N-closed, then any convergent net in B must converge to a pointin B.

(ii) In T − NEAR, if the limit of any convergent net in B always remainsin B, then B is N-closed.

64 Z. Yang

The next several theorems show that the properties of being totally bounded,compact or N2 are productive respectively.

Theorem 4.13. Let X be a set, (Y, η) be a N-space, and f : X → Y be aN-preserving map. Then f−1(η) is totally bounded if and only if η is totallybounded.

Proof. Assume that f−1(η) is totally bounded. We would like to show that η istotally bounded by showing that it meets condition (B1). Arbitrarily take a B

such that ηB. Then f−1(B) /∈ f−1(η). There should be a finite subcollectionof B, say B0 ⊆ B, such that

⋂

f−1(B0) = φ. Hence⋂

B0 = φ.Now we assume that η is totally bounded. Arbitrarily take A /∈ f−1(η),

then ηf(A ). So there should be a finite subcollection of A , say A0 ⊆ A , suchthat

⋂

f(A0) = φ. Now we can easily see that⋂

A0 = φ.

Theorem 4.14. Let ξα : α ∈ Λ be a family of nearness structures on X.Let ξ = supξα : α ∈ Λ. Then ξ is totally bounded if and only if for eachα ∈ Λ, ξα is totally bounded.

Proof. First we assume that for each α ∈ Λ, ξα is totally bounded. Then for anyξA , there should be finitely many ξαi

, i = 1, 2, ...n as well as Aαi, i = 1, 2, ...n

such that Aαi/∈ ξαi

and A ≺n∨

i=1Aαi

. Since each ξαiis totally bounded, each

Aαicontains a finite subcollection Bαi

and⋂

Bαi= φ.

n∨

i=1Bαi

is a finite

subcollection ofn∨

i=1Aαi

. We will claim that⋂ n

∨i=1

Bαi= ∅. Take an arbitrary

point x ∈ X , then for each i = 1, 2, ..., n, there is a Bxi ∈ Bαi

such that x /∈ Bxαi

.

So x /∈n∪

i=1Bx

αi. Hence x /∈

⋂ n∨

i=1Bαi

. This shows that⋂ n

∨i=1

Bαi= φ. So ξ is

totally bounded.Now we assume that ξ is totally bounded. Arbitrarily take a ξα. Then for

any ξαAα, we have ξAα. Since ξ is assumed to be totally bounded, Aα musthave finite subcollection with empty intersection.

Theorem 4.15. If ξα : α ∈ Λ is a family of nearness structures on X . Itsproduct (

∏

α∈Λ Xα,∏

α∈Λ ξα) is totally bounded if and only if each (Xα, ξα) is

totally bounded. Particularly, if (Y, η) is totally bounded and F ⊆ Y X whereX is a non-empty set, then (F , ξρ) is totally bounded.

Proof. The first conclusion can be deduced from Theorem 4.13 and Theo-rem 4.14. The second conclusion can be deduced from the Theorem 3.18.

The next Lemma, due to Herrlich ([8], 4.5 Proposition, Part (2)), will beused in the proof of the following theorem.

Lemma 4.16. For a T-N space, the following conditions are equivalent:

(1) (X, ξ) is contigual;(2) (X, ξ) is totally bounded;(3) (X, ξ) is compact.


Theorem 4.17. If ξα : α ∈ Λ is a family of nearness structures on X. Ifeach (Xα, ξα) is compact, then its product (

∏

α∈Λ Xα,∏

α∈Λ ξα) is also compact.

Proof. Recall that a compact nearness space satisfies condition (T) and (C).It is easy to verify that the product is a T-N space, if each (Xα, ξα) is a T-Nspace. According to Lemma 4.16, a T-N space is C-N if and only it it is totallybounded. So the conclusion of this theorem follows from Theorem 4.15.

Theorem 4.18. If ξα : α ∈ Λ is a family of nearness structures on X . Ifeach (Xα, ξα) is N2, then the product (X, ξ) = (

∏

α∈Λ Xα,∏

α∈Λ ξα) is also N2.

Proof. By the definition of product of nearness structures, ξ = supP−1α (ξα) :

α ∈ Λ, where Pα :∏

α∈Λ Xα → Xα are natural projection maps. Let x, y ∈ Xand x 6= y, there should be at least one α ∈ Λ such that Pα(x) 6= Pα(y). Sinceξα is N2, there are Aα ⊆ X and Bα ⊆ X such that Aα ∩ Bα = ∅ and

Pα(x) ∈ Xα − clξα(Xα − Aα), Pα(y) ∈ Xα − clξα

(Xα − Bα).

It is easy to see that P−1α (Aα) ∩ P−1

α (αB) = ∅. We will try to show that

x ∈ X − clξ(X − P−1α (Aα)),

andy ∈ X − clξ(X − P−1

α (Bα)).

Take an arbitrary point

z ∈ P−1α (Xα − clξα

(Xα − Aα)),

thenPα(z) ∈ Xα − clξα

(Xα − Aα).

andPα(z), Xα − Aα /∈ ξα.

P−1α (Pα(z)), P−1

α (Xα − Aα) /∈ P−1α (ξα).

z, X − P−1α (Aα) /∈ P−1

α (ξα).

The last statement is true since

z ∈ P−1α (Pα(z))

andP−1

α (Xα − Aα) ⊆ X − P−1α (Aα).

Therefore,z ∈ X − clP−1

α (ξα)(X − P−1α (Aα)).

This shows that

P−1α (Xα − clξα

(Xα − Aα)) ⊆ X − clP−1

α (ξα)(X − P−1α (Aα)).

Hence

x ∈ P−1α (Xα − clξα

(Xα − Aα))

⊆ X − clP−1

α (ξα)(X − P−1α (Aα))

⊆ X − clξ(X − P−1α (Aα)).

66 Z. Yang

By similar argument,

y ∈ P−1α (Xα − clξα

(Xα − Bα))

⊆ X − clP−1

α (ξα)(X − P−1α (Bα))

⊆ X − clξ(X − P−1α (Bα)).

Therefore, (X, ξ) is a N2- space.

Now we will try to establish the relation between the compactness of theunderderlining set Y and a function space F ⊆ Y X .

Theorem 4.19. Let X be a set, and (Y, η) be a compact N-space. F ⊆ Y X .Then

(i) The condition (a) is sufficient for (F , ξρ) to be compact.(ii) If (Y, η) is also N2, then the condition (a) is also necessary for (F , ξρ)

to be compact.

(a) F is N-closed in (Y X , ξ∗).

Proof. (i) Since (Y, η) is a compact N-space. By Theorem 4.17, (Y X , ξ∗)is compact. By Theorem 3.12, F ⊆ Y X , as an N-closed subset of acompact space, is also compact under the subspace nearness structure.

(ii) If (Y, η) is a N2-space. By Theorem 4.18, (Y X , ξ∗) is an N2- space.Then by Theorem 3.17, (F , ξρ), as a compact subspace of an N2- space,is N-closed.

Theorem 4.20. If X is a set, and (Y, η) is an N-space. F ⊆ Y X . Then

(i) the conditions (a) and (b) are sufficient for (F , ξρ) to be compact.(ii) If (Y, η) is also N2, then the conditions (a) and (b) are also necessary

for (F , ξρ) to be compact.

(a) F is N-closed in (Y X , ξ∗)(b) For any x ∈ X, F [x] = f(x) : f ∈ F is contained in a compact

subspace of (Y, η).

Proof. (i) Assume that (Yx, ηx) is a compact nearness subspace of (Y, η)with ηx = η|Yx

and F [x] ⊆ Yx ⊆ Y . Then F ⊆ (∏

x∈X Yx,∏

x∈X ηx)and the later space is compact, according to Theorem 4.17. It is easyto see that ξρ =

∏

x∈X ηx|F . Since F is N-closed also, it is N-compactdue to Theorem 3.12.

(ii) If (Y, η)is a N2-space, and (F , ξρ) is compact. It follows from Theo-rem 3.17 that (a) is true. And since the evaluation map ex : (F , ξρ) →(Y, η) is N-preserving. By Lemma 3.14, F [x] = f(x) : f ∈ F =ex(f) : f ∈ F is also compact.


Summary. This paper essentially lays the foundation for some possible ap-plications of the theory of nearness function spaces in digital topology. Themain results is the introduction of the pointwise convergent nearness spacesin the function spaces in such a way that is consistent with the existing andestablished structures. Two Ascoli’s type of theorems on nearness spaces areestablished also.

Any deformation of a digital image (such as thinning) can be considered asa function from the digital plane to itself. Of course, we would prefer thosefunctions to preserve some properties of the digital image, such as ”nearness”.When a sequence of deformations are applied to a digital image, we would liketo be able to make some type of projection or prediction about the final imagesbased on the type of deformations involved in the sequence. We believe that theline of work presented in this paper will be helpful to address those issues.

References

[1] H. L. Bentley, J. W. Carlson and H, Herrlich, On the epireflective Hull of Top in Near,Topology Appl. 153 (2007), 3071-3084.

[2] H. L. Bentley and H. Herrlich, Compleness Is Productiver, Categorical Topology (Dold A.and Eckmann B., Eds.), 719 Proceedings, Berlin , Springer-Verlag, Berlin, Heidelberg,New York (1978), 13–17.

[3] H. L. Bentley and H. Herrlich, Ascoli’s Theorem for a Class of Merotopic spaces, Con-

vergence Structures (Dold A. and Eckmann B. Eds.), Proc. Bechyne Conf. , Springer-Verlag, Berlin, Heidelberg, New York (1984), 47–54.

[4] B. B. Chaudhuri, A new definition of neighborhood of a point in multi-dimensional

space, Pattern Recognition Letters 17 (1996), 11–17.[5] P. R. Fu, Proximity on Function Space of Set-Valued Functions, Journal of Mathematics

13, no. 3 (1993), 347–350.[6] J. W. Gray, Ascoli’s Theorem for topological categories, Categorical Aspects of Topology

and Analysis (Dold A. and Eckmann B. Eds.), 915 Proceedings, Berlin , Springer-Verlag,Berlin, Heidelberg, New York (1978), 86–104.

[7] N. C. Heldermann, Concentrated Nearness Spaces, Categorical Topology (Dold A. andEckmann B., Eds.), 719 Proceedings, Berlin , Springer-Verlag, Berlin, Heidelberg, NewYork (1978), 122–136.

[8] H. Herrlicht, A Concept of Nearness, General Topology and Application 5 (1974), 191–212.

[9] W. N. Hunsaker and P. L. Sharma, Nearness Structure Compatible with a Topological

Space, Arch. Math. XXV (1974), 172–178.[10] V. M. Ivanova and A. A. Ivanov, Contiguity spaces and bicompact extensions, Izv. Akad.

Nauk. SSSR 23 (1959), 613–634.[11] J. L. Kelley, General Topology, (D. Van Norstrand, Princeton Toronto London New

Work, 1974).[12] L. Latecki and F. Prokop, Semi-proximity continuous functions in digital images, Pat-

tern Recognition Letters 16 (1995), 1175–1187.[13] Y. F. Lin and D. Rose, Ascoli’s Theorem for Spaces of Multifunctions, Pacific Journal

of Mathematics 34, no. 3 (1970), 741–747.[14] S. A. Naimpally and B. D. Warrack, Proximity Spaces, (Cambridge University Press,

London 1970).[15] J. F. Peters, A. Skowron and J. Stepaniuk, Nearness of Objects: Extension of Approxi-

mation Space Model, Fundamenta Informaticae 79 (2007), 497–512.

68 Z. Yang

[16] P. Ptak and W. G. Kropatsch, Nearness in Digital Images and Proximity Spaces, DGCI2000, LNCS 1953 (2000), 69–77.

[17] G. Sonck, An Ascoli Theorem for Sequential Spaces, Int. J. Math. Math. Sci. 26, no. 5(2001), 303–315.

[18] M. Wolski, Approximation Spaces and Nearness Structures, Fundamenta Informaticae79 (2007), 567–577.

[19] Z. Yang, A New Proof on Embedding the Category of Proximity Spaces into the Category

of Nearness Spaces, Fundamenta Informaticae 88, no. 1-2 (2008), 207–223.[20] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, The Joy

of Cats, (John Wiley and Sons, New Work 1970).

Received May 2008

Accepted October 2008

Zhanbo Yang ([email protected])Department of Mathematical Sciences, University of the Incarnate Word, 4301Broadway, San Antonio, TX 78209, USA



Volume 10, No. 1, 2009

pp. 69-83

F-supercontinuous functions

J. K. Kohli, D. Singh∗ and Jeetendra Aggarwal

Abstract. A strong variant of continuity called ‘F -supercontinuity’is introduced. The class of F -supercontinuous functions strictly con-tains the class of z-supercontinuous functions (Indian J. Pure Appl.Math. 33 (7) (2002), 1097–1108) which in turn properly contains theclass of cl-supercontinuous functions (≡ clopen maps) (Appl. Gen.Topology 8 (2) (2007), 293–300; Indian J. Pure Appl. Math. 14 (6)(1983), 762–772). Further, the class of F -supercontinuous functions isproperly contained in the class of R-supercontinuous functions whichin turn is strictly contained in the class of continuous functions. Basicproperties of F -supercontinuous functions are studied and their placein the hierarchy of strong variants of continuity, which already exist inthe mathematical literature, is elaborated. If either domain or range isa functionally regular space (Indagationes Math. 15 (1951), 359–368;38 (1976), 281–288), then the notions of continuity, F-supercontinuityand R-supercontinuity coincide.

2000 AMS Classification: 54C08, 54C10, 54D10, 54D20, 54D30

Keywords: z-supercontinuous function, F-supercontinuous function, func-tionally regular space, functionally Hausdorff space, F-completely regular space,F-quotient topology

1. Introduction

Several strong variants of continuity occur in the lore of mathematical liter-ature which arise in many branches of mathematics and applications of math-ematics.

In many situations in topology, analysis and other disciplines continuity isnot sufficient and a strong form of continuity is required to meet the demand ofa particular situation. The strong variants of continuity with which we shall bedealing in this paper include, among others, are strongly continuous functions[16], perfectly continuous functions [20], clopen maps [21] (≡ cl-supercontinuous

∗This research was partially supported by University Grants Commission, India.

70 J. K. Kohli, D. Singh and J. Aggarwal

functions [23]), z-supercontinuous functions [8], D-supercontinuous functions[10], D∗-supercontinuous functions [22], Dδ-supercontinuous functions [11],strongly θ-continuous functions [19] and supercontinuous functions [18]. Themain purpose of this paper is to introduce a new class of functions called ‘F -supercontinuous functions’, study their basic properties and discuss their placein the hierarchy of strong variants of continuity that already exist in the math-ematical literature. The notion of F -supercontinuous functions arise naturallyin case either domain or range is a functionally regular space ([1], [2]). It turnsout that the class of F -supercontinuous functions properly includes the classof z-supercontinuous functions [8] and is strictly contained in the class of R-supercontinuous functions [14] which in turn is properly contained in the classof continuous functions. Further if either domain or range is a functionallyregular space ([1], [2]) then all the three classes of (i) F -supercontinuous func-tions (ii) R-supercontinuous functions, and (iii) continuous functions coincide.Moreover, if either X or Y is a completely regular space, then all these threeclasses of functions are identical with the class of z-supercontinuous functions[8]. Furthermore, if either domain or range is zero dimensional space, then allthe four above classes of functions coincide with the class of cl-supercontinuousfunctions ([21], [23]).

Section 2 is devoted to the preliminaries and basic definitions. In Section 3,we introduce the notion of ‘F -supercontinuous function’ and elaborate on itsplace in the hierarchy of strong variants of continuity which already exist inthe literature. Basic properties of F -supercontinuous functions are studied inSection 4, while properties of graph of an F -supercontinuous function are dis-cussed in Section 5. Interplay of topological properties and F-supercontinuousfunctions is investigated in Section 6 and the notion of F -quotient topology isformulated in Section 7. Change of topology of a topological space (X, τ) intoa functionally regular topology τF and a completely regular topology τz areconsidered in Section 8 wherein interrelations betweenτ , τF and τz are elab-orated and alternative proofs of certain results in the preceding sections aresuggested.

2. Basic definitions and preliminaries

A collection β of subsets of a space X is called as open complementary system[6] if β consists of open sets such that for every B ∈ β, there exist B1, B2, . . . ∈ βwith B = ∪X \ Bi : i ∈ N. A subset A of a space X is called a stronglyopen Fσ-set [6] if there exists a countable open complementary system β(A)with A ∈ β(A). The complement of a strongly open Fσ-set is called stronglyclosed Gδ-set. A subset A of a space X is called a regular Gδ-set [17] if A isan intersection of a sequence of closed sets whose interiors contain A, i.e., if

A =∞⋂

n=1

Fn =∞⋂

n=1

F n , where each Fn is a closed subset of X . The complement

of a regular Gδ-set is called a regular Fσ-set. An open subset A of a space Xis said to be r-open [14] if it is expressible as a union of closed sets.

F-supercontinuous functions 71

Definition 2.1. A function f : X → Y from a topological space X into atopological space Y is said to be

(a) strongly continuous [16] if f(A) ⊂ f(A) for each subset A of X.(b) perfectly continuous [20] if f−1(V ) is clopen in X for every open set

V ⊂ Y .(c) cl-supercontinuous [23] (≡ clopen map [21]) if for each x ∈ X and each

open set V containing f(x) there is a clopen set U containing x such thatf(U) ⊂ V .

(d) z-supercontinuous [8] if for each x ∈ X and for each open set V con-taining f(x), there exists a cozero set U containing x such that f(U) ⊂ V .

(e) Dδ-supercontinuous [11] if for each x ∈ X and for each open set Vcontaining f(x), there exists a regular Fσ set U containing x such thatf(U) ⊂ V .

(f) D-supercontinuous [10] if for each x ∈ X and each open set U containingf(x) there exists an open Fσ-set V containing x such that f(V ) ⊂ U .

(g) D∗-supercontinuous [22] if for each x ∈ X and each open set U con-taining f(x) there exists a strongly open Fσ-set V containing x such thatf(V ) ⊂ U .

(h) strongly θ-continuous [19] if for each x ∈ X and for each open set Vcontaining f(x), there exists an open set U containing x such that f(U) ⊂V .

(i) R-supercontinuous [14] if for each x ∈ X and each open set U containingf(x) there exists an r-open set V containing x such that f(V ) ⊂ U .

(j) supercontinuous [18] if for each x ∈ X and for each open set V containingf(x), there exists a regular open set U containing x such that f(U) ⊂ V .

Definition 2.2. A topological space X is said to be

(i) functionally regular ([1], [2]) if for each closed set A and a point x /∈ Athere exists a continuous real-valued function f defined on X such thatf(x) /∈ f(A); or equivalently for each x ∈ X and each open set U con-taining x there exists a zero set Z such that x ∈ Z ⊂ U .

(ii) D-regular space [6] if it has a base of open Fσ-sets.(iii) functionally Hausdorff [25] If for x, y ∈ X, x 6= y there exists a

continuous function f : X → [0, 1] such that f(x) 6= f(y).(iv) countably H-closed if it is Hausdorff and every countable open cover of

X has a finite subcollection whose union is dense in X.(v) semiregular if it has a base of regular open sets.

In order to systematize the study of separation by continuous real-valuedfunctions, Van Est and Freudenthal [25] introduced the notion of a function-ally regular space and showed its distinctiveness from the standard separationaxioms and other separation axioms defined by them. Further properties offunctionally regular spaces have been studied by Aull ([1], [2]).


3. F -supercontinuous functions

An open set U in a space X is said to be F -open if for each x ∈ U , thereexists a zero set Z in X such that x ∈ Z ⊂ U , or equivalently, U is expressibleas a union of zero sets. The complement of an F -open set will be referred toas an F -closed set.

Definition 3.1. A function f : X → Y from a topological space X into atopological space Y is said to be F -supercontinuous if for each x ∈ X andeach open set U containing f(x) there exists an F -open set V containing x suchthat f(V ) ⊂ U .

The following diagram reflects upon the place of F -supercontinuous func-tions in the hierarchy of strong variants of continuity that already exist in theliterature. The implications are either well known or immediately follow fromdefinitions.

However, none of the above implications is reversible which is either wellknown or follows from the following observations and examples.

Observations and Examples

3.2 If either X or Y is a functionally regular space, then every continuousfunction f : X → Y is F -supercontinuous and hence R-supercontinuous.

3.3 If either X or Y is a completely regular space, then every continuous func-tion f : X → Y is z-supercontinuous.


3.4 Let X = Y be the regular space due to Hewitt [7] on which every continuousreal valued function is constant and let f denote the identity map defined onX . Then f is strongly θ-continuous and so R-supercontinuous but it is notF -supercontinuous.

3.5 Let X be a functionally regular space which is not completely regular andlet Y = X . Then the identity mapping defined on X is F -supercontinuous butnot z-supercontinuous.

3.6 Let X be the space of [3, Exercise 24, p. 139]. Then X is a Hausdorffsemiregular space which is not regular. Further, Aull pointed out that thespace X is a functionally regular space (see [1, Example 3]). Let f denote theidentity mapping defined on X . Then the function f is supercontinuous as wellas F -supercontinuous but not strongly θ-continuous.

3.7 Let us denote by X the space of Arens square [24, Example 80, p. 98].Then X is a Hausdorff space which is not functionally Hausdorff and hence nota functionally regular space. Also, X is semiregular but not regular. So theidentity mapping defined on X is supercontinuous but not F -supercontinuous.

3.8 Let us denote by X the space of irregular lattice topology [24, Exam-ple 79, p. 97]. Then X is a functionally Hausdorff Lindelof space which is nota semiregular space. In view of [1, Theorem 3], the space X is a functionallyregular space. Let f denote the identity mapping defined on X . Then f is anF -supercontinuous function but not supercontinuous.

3.9 Let us denote by X the real line with the smallest topology generated bythe Euclidean topology and the cocountable topology on X . The space X isa functionally regular space, since it is a functionally Hausdorff Lindelof space(see [1, Theorem 3]). The space X is not D-regular, since X is not a subpara-compact space and every Lindelof, D-regular space is subparacompact (see [4,Theorem 2]). Then the identity mapping defined on X is an F -supercontinuousfunction but not a D-supercontinuous function.

Proposition 3.10. Let f : X → Y be a continuous function, defined on afunctionally Hausdorff Lindelof space X. Then f is F -supercontinuous.

Proof. Since a functionally Hausdorff Lindelof space is functionally regular (see[1]) and since every continuous function defined on a functionally regular spaceis F -supercontinuous, f is F -supercontinuous.

Proposition 3.11. Let f : X → Y be a continuous function. If X is acountably paracompact functionally regular space, then f is F -supercontinuousas well as strongly θ-continuous.

Proof. Since every continuous function defined on a functionally regular spaceis F -supercontinuous, so is f . Again, since every countably paracompact func-tionally regular space is regular (see [1]), and since every continuous functiondefined on a regular space is strongly θ-continuous, f is strongly θ-continuous.


Proposition 3.12. Let f : X → Y be a continuous function defined on acountably compact functionally regular space X. Then f is z-supercontinuous.

Proof. This is immediate in view of the fact that every countably compactfunctionally regular space is completely regular (see [1]), and every continuousfunction defined on a completely regular space is z-supercontinuous.

Proposition 3.13. Let f : X → Y be a continuous function defined on acountably H-closed, semiregular, functionally regular space X. Then f is z-supercontinuous.

Proof. This is immediate from the fact that a countably H-closed, semiregular,functionally regular space X is completely regular (see [1]).

4. Basic properties of F-supercontinuous functions

Theorem 4.1. For a function f : X → Y from a topological space X into atopological space Y , the following statements are equivalent:

(a) f is F -supercontinuous.(b) The inverse image of every open subset of Y is F -open in X.(c) The inverse image of every closed subset of Y is F -closed in X.(d) The inverse image of every subbasic open subset of Y is F -open in X.

Proof. It is easy using definitions.

Definition 4.2. Let X be a topological space and let A ⊂ X. A point x ∈ Xis said to be an F-adherent point of the set A if every F -open set containingx has non-empty intersection with A. Let AF denote the set of all F -adherentpoints of the set A. The set A is F -closed if and only if A = AF .

Theorem 4.3. For a function f : X → Y the following statement are equiva-lent.

(a) f is F -supercontinuous.

(b) f(AF ) ⊂ f(A) for every A ⊂ X.(c) (f−1(B))F ⊂ f−1(B) for every B ⊂ Y .

Proof. (a) ⇒ (b). Since f(A) is closed in Y , by Theorem 4.1 f−1(f(A)) is an F -

closed set in X . Again, since A ⊂ f−1(f(A)), AF ⊂ [f−1(f(A))]F = f−1(f(A))

and so f(AF ) ⊂ f(f−1(f(A))) ⊂ f(A).

(b) ⇒ (c). Let B ⊂ Y . Then f((f−1(B))F ) ⊂ f(f−1(B)) ⊂ B and so it followsthat (f−1(B))F ⊂ f−1(B).(c) ⇒ (a). Let B be any closed set in Y . Then (f−1(B))F ⊂ f−1(B). Since

f−1(B) ⊂ f−1(B) ⊂ (f−1(B))F , f−1(B) = (f−1(B))F which in turn impliesthat f is F -supercontinuous.

Definition 4.4. A filterbase F is said to F-converge to a point x, written as

FF→ x if every F -open set containing x contains a member of F .

Theorem 4.5. A function f : X → Y is F -supercontinuous if and only if foreach x ∈ X and each filter base F in X that F -converges to x, f(F) → f(x).


Proof. Suppose that f is F -supercontinuous and let F be a filter base in Xthat F -converges to x. To show that the filter base f(F) converges to f(x),let W be any open set containing f(x). Then x ∈ f−1(W ) and f−1(W ) isF -open. Since the filter base F converge to x, there exists F ∈ F such thatF ⊂ f−1(W ). Then f(F ) ⊂ f(f−1(W )) ⊂ W and so f(F) → f(x).

Conversely, let W be an open set containing f(x). Now, the filter F gener-ated by the filterbase Bx consisting of F -open sets containing x, F -convergesto x. Since by hypothesis f(F) → f(x), there exists a member f(F ) of f(F)such that f(F ) ⊂ W . Choose B ∈ Bx such that B ⊂ F . Since B is an F -openset containing x and since f(B) ⊂ f(F ) ⊂ W , f is F -supercontinuous.

Theorem 4.6. If f : X → Y is F -supercontinuous and g : Y → Z is con-tinuous, then the composition g f is F -supercontinuous. In particular, thecomposition of F -supercontinuous functions is F -supercontinuous.

In general F -supercontinuity of g f need not imply even continuity of f .For example, let X be the real line with cofinite topology, Y be the real linewith cocompact topology and Z be the real line with indiscrete topology. Letf : X → Y and g : Y → Z be the identity mappings. Then g f and g areF -supercontinuous. However, f is not continuous.

It is routine to verify that F -supercontinuity is invariant under restrictionsand enlargement of range.

Definition 4.7. A function f : X → Y is said to be F-open (F-closed) iff(A) is open (closed) in Y for every F -open (F -closed) set A in X.

Theorem 4.8. Let f : X → Y be an F -open (F -closed), F -supercontinuoussurjection and g : Y → Z be any function. Then the composition g f is F -supercontinuous if and only if g is continuous. Further, if in addition f maps F -open (F -closed) set to F -open (F -closed) set, then g is an F -supercontinuousfunction.

Proof. Suppose that g f is F -supercontinuous. To show that g is continuous,let W be an open (closed) subset of Z. Then by Theorem 4.1 (g f)−1(W ) =f−1(g−1(W )) is F -open (F -closed) in X . Since f is an F -open (F -closed) sur-jection f(f−1(g−1(W ))) = g−1(W ) is open (closed) in Y and so g is continuous.

Conversely, suppose that g is continuous and let W be an open (closed)set in Z. Then g−1(W ) is open (closed) in Y . Since f is F -supercontinuous,f−1(g−1(W )) = (g f)−1(W ) is F -open (F -closed) in X and so g f is F -supercontinuous.

For the last assertion we need only note that F -openness (F -closedness) ofg−1(W ) ensures the F -supercontinuity of g.

Theorem 4.9. Let f : X → Y be any function. Then the following statementsare true.

(a) If Uα : α ∈ Λ is an F -open cover of X and for each α, fα = f |Uα isF -supercontinuous, then f is F -supercontinuous.


(b) If Fi : i = 1, . . . , n is an F -closed cover of X and fi = f |Fi is F -supercontinuous, then f is F -supercontinuous.

Proof. (a) Let V be any F -open set in Y . Then f−1(V ) = ∪f−1α (V ) : α ∈ Λ.

Since each fα is F -supercontinuous, In view of Theorem 4.1 each f−1α (V ) is F -

open in Uα and hence in X . Since any union of F -open sets is F -open, f−1(V )is F -open.

(b) Let B be any F -closed set in Y . Then f−1(B) =n⋃

i=1

f−1i (B). Since each

fi is F -supercontinuous, by Theorem 4.1 each f−1i (B) is F -closed in Fi and

hence in X . Then f−1(B) being a finite union of F -closed sets is F -closed. Sof is F -supercontinuous.

Theorem 4.10. Let fα : X → Xα : α ∈ Λ be a family of functions and letf : X →

∏α∈Λ Xα be defined by f(x) = (fα(x)) for each x ∈ X. Then f is

F -supercontinuous if and only if each fα : X → Xα is F -supercontinuous.

Proof. Let f : X →∏

α∈Λ Xα be F -supercontinuous. Then the composition

pα f = fα, where pα denotes the projection of∏

α∈Λ Xα onto αth-coordinatespace Xα. So in view of Theorem 4.6 each fα is F -supercontinuous.

Conversely, suppose that each fα : X → Xα is F -supercontinuous. Toshow that the function f is F -supercontinuous, it is sufficient to show thatf−1(V ) is F -open for each open set V in the product space

∏α∈Λ Xα. Since

arbitrary unions and finite intersections of F -open sets is F -open, it sufficesto prove that f−1(S) is F -open for every subbasic open set S in the productspace

∏α∈Λ Xα. Let Vβ ×

∏α6=β Xα be a subbasic open set in

∏α∈Λ Xα. Then

f−1(Vβ ×∏

α6=β Xα) = f−1(p−1β (Vβ)) = f−1

β (Vβ) is F -open in X . Hence f isF -supercontinuous.

Theorem 4.11. For each α ∈ ∆, let fα : Xα → Yα be a mapping and letf :

∏Xα →

∏Yα be a mapping defined by f((xα)) = (fα(xα)) for each (xα)

in∏

Xα. Then f is F -supercontinuous if and only if fα is F -supercontinuousfor each α ∈ ∆.

Proof. Let f :∏

Xα →∏

Yα be F -supercontinuous. Let Vβ be an open sub-set of Yβ . Then Vβ × (

∏α6=β Yα) is a subbasic open subset of the product

space∏

Yα. Since f is F -supercontinuous, f−1(Vβ ×∏

α6=β Yα) = f−1β (Vβ) ×

(∏

α6=β Xα) is F -open in∏

Xα. Consequently, f−1β (Vβ) is a F -open set in Xβ

and hence fβ is a F -supercontinuous.Conversely, suppose that each fα : Xα → Yα is F -supercontinuous. Let

V = Vβ × (∏

α6=β Yα) be a subbasic open set in∏

Yα. Since each fα is F -

supercontinuous, and since f−1(V ) = f−1(Vβ × (∏

α6=β Yα)) = f−1β (Vβ) ×

(∏

α6=β Xα), f−1(V ) is F -open, and so f is F -supercontinuous.


Theorem 4.12. Let f : X → Y be a function and g : X → X × Y , definedby g(x) = (x, f(x)) for each x ∈ X, be the graph function. Then g is F -supercontinuous if and only if f is F -supercontinuous and X is functionallyregular.

Proof. To prove necessity, suppose that g is F -supercontinuous. Then thecomposition f = py g is F -supercontinuous, where py is the projection fromX × Y onto Y . Let U be any open set in X and let x ∈ U . Then U × Y is anopen set containing g(x). Since g is F -supercontinuous, there exists an F -openset Wx containing x such that g(Wx) ⊂ U ×Y . Thus x ∈ Wx ⊂ U and since Uis a union of F -open sets, then it is F -open and so X is functionally regular.

To prove sufficiency, let x ∈ X and let W be an open set containing g(x).There exist open sets U ⊂ X and V ⊂ Y such that (x, f(x)) ∈ U × V ⊂ W .Since X is functionally regular, there exists an F -open set G1 in X containingx such that x ∈ G1 ⊂ U . Since f is F -supercontinuous, there exists an F -openset G2 in X containing x such that f(G2) ⊂ V . Let G = G1 ∩ G2. Then G isan F -open set containing x and g(G) ⊂ U × V ⊂ W , which implies that g isF -supercontinuous.

The following example shows that the hypothesis that ‘X is functionallyregular’ in Theorem 4.12 cannot be omitted.

Example 4.13. Let X = Y = a, b, c, d. Let the topology on X be givenby τ = φ, X, a, b, d, a, b, d and let Y be equipped with indiscrete topol-ogy. Let f : X → Y be the constant function which takes the value b. Thenf is F -supercontinuous but the graph function g : X → X × Y is not F -supercontinuous.

Theorem 4.14. Let f, g : X → Y be F -supercontinuous functions from Xinto a Hausdorff space Y . Then the equalizer E = x ∈ X : f(x) = g(x) ofthe functions f and g is an F -closed set in X.

Proof. To show that E is F -closed, we shall show that its complement X \E isan F -open subset of X . Let x ∈ X\E. Then f(x) 6= g(x). Since Y is Hausdorff,there exist disjoint open sets V and W containing f(x) and g(x), respectively.Since f and g are F -supercontinuous, f−1(V ) and g−1(W ) are F -open setscontaining x. Then U = f−1(V )∩g−1(W ) is an F -open set containing x whichis contained in X \ E and so X \ E is F -open.

Corollary 4.15. Let X be a Hausdorff space. Then the set of fixed points ofevery F -supercontinuous function f : X → X is an F -closed set.

Definition 4.16. A space X is said to be F -completely regular if for everyF -closed set A and a point x outside A there exists a continuous functionf : X → [0, 1] such that f(x) = 0 and f(A) = 1.

Theorem 4.17. Let f : X → Y be an F -supercontinuous function. If X isF -completely regular, then f is z-supercontinuous.


Proof. Let x ∈ X and let V be an open set containing f(x). Since f is F -supercontinuous, there exists an F -open set U containing x such that f(U) ⊂V . Since X is a F -completely regular space, there exists a continuous functionh : X → [0, 1] such that h(x) = 0 and h(X \U) = 1. Then h−1[0, 1) is a cozeroset containing x and contained in U and so it is mapped into V by f . Thisshows that f is z-supercontinuous.

5. Properties of the graph of an F -supercontinuous function

Let f : X → Y be an F -supercontinuous function. Since every F -superconti-nuous function is continuous, the family 1x, f, where 1x denotes the identitymapping on X , separates points and separates points from closed sets. There-fore, the mapping g : X → X ×Y defined by g(x) = (x, f(x)) is an embeddingof X into X × Y . Thus X is homeomorphic to its graph G(f) = g(x) and soevery topological property enjoyed by X is also enjoyed by its graph G(f).

The next two notions reflect upon the fact that how the graph G(f) of anF -supercontinuous function f : X → Y is situated in the product space X×Y .

Definition 5.1. Let f : X → Y be a function from a topological space X intoa topological space Y . The graph G(f) of f is said to be

(i) F-closed with respect to X if for each (x, y) /∈ G(f), there exist opensets U and V containing x and y, respectively such that U is F -open and(U × V ) ∩ G(f) = φ.

(ii) F-closed with respect to X× Y if for each (x, y) /∈ G(f), there existF -open sets U and V containing x and y, respectively such that (U ×V )∩G(f) = φ.

Proposition 5.2. For a topological space X the following are equivalent.

(a) X is functionally Hausdorff.(b) Every pair of distinct points in X are contained in disjoint cozero sets.(c) Every pair of distinct points in X are contained in disjoint F -open sets.

Proof. The implication (a)⇒(b)⇒(c) are trivial. To prove (c)⇒(a), let x, y ∈X , x 6= y and let U and V be disjoint F -open sets containing x and y, re-spectively. Let A and B be the zero sets in X such that x ∈ A ⊂ U andy ∈ B ⊂ V . Let f, g be the real-valued functions defined on X such thatZ(f) = A and Z(g) = B, where Z(f) and Z(g) denote the zero sets of f and g,respectively. Let h : X → R be the function defined by h(t) = f(t)/[f(t)+g(t)],for t ∈ X . Then h is a continuous function defined on X such that h(x) = 0and h(y) = 1.

Theorem 5.3. If f : X → Y is F -supercontinuous and Y is functionallyHausdorff, then G(f), the graph of f is F -closed with respect to X × Y .

Proof. Let x ∈ X and let y 6= f(x). Since Y is functionally Hausdorff, thereexist disjoint F -open sets V and W containing y and f(x), respectively. ByF -supercontinuity of f there exists an F -open set U containing x such that


f(U) ⊂ W ⊂ Y \ V . Consequently, U × V contains no point of G(f). HenceG(f) is F -closed with respect to X × Y .

The following result is immediate.

Proposition 5.4. If f : X → Y is F -supercontinuous and Y is Hausdorff,then G(f), the graph of f is F -closed with respect to X.

6. Topological properties and F -supercontinuity

Theorem 6.1. Let f : X → Y be an F -supercontinuous open bijection. ThenX and Y are homeomorphic functionally regular spaces.

Proof. Let U be an open set in X and let x ∈ U . Since f is an openmap, f(U) is an open set containing f(x). Since f is F -supercontinuous,there exists an F -open set G containing x such that f(G) ⊂ f(U). Now,x ∈ f−1(f(G)) ⊂ f−1(f(U)). Since f is a bijection, f−1(f(G)) = G andf−1(f(U)) = U . Thus x ∈ G ⊂ U . So U being a union of F -open sets is F -open. Thus X is a functionally regular space. Since f is a homeomorphism andfunctional regularity is a topological property, Y is functionally regular.

Theorem 6.2. Let f : X → Y be an F -supercontinuous injection into a T0-space, then X is a functionally Hausdorff space.

Proof. Let x and y be two distinct points in X . Then f(x) 6= f(y). SinceY is a T0-space, there exists an open set V containing one of the points f(x)or f(y) but not the other. To be precise, assume that f(x) ∈ V . Since fis F -supercontinuous, f−1(V ) is an F -open set containing x but not y. Sothere exists a zero set Zx such that x ∈ Zx ⊂ f−1(V ). Let ϕ : X → [0, 1]be the continuous function such that Z(ϕ) = Zx the zero set of ϕ. Thenϕ(x) = 0 6= ϕ(y). Let G and H be disjoint open sets in [0,1] containing ϕ(x)and ϕ(y) respectively. It follows that ϕ−1(G) and ϕ−1(H) are disjoint cozerosets in X containing x and y, respectively. So X is a functionally Hausdorffspace.

Corollary 6.3. Let f : X → Y be a z-supercontinuous injection into a T0-space, then X is a functionally Hausdorff space.

Definition 6.4 ([13]). A space X is said to be F -compact if every F -opencover of X has a finite subcover.

Theorem 6.5. If f : X → Y is an F -supercontinuous surjection from anF -compact space X onto Y , then Y is compact.

Proof. Let β = Vα|α ∈ ∆ be an open cover of Y . In view of F -supercontinuityof f , f−1(Vα) : α ∈ ∆ is an F -open cover of X . Since X is F -compact, there

exists a finite subset α1, . . . , αn of ∆ such thatn⋃

i=1

f−1(Vαi) = X . Since f is

surjection, Vα1, . . . , Vαn

is a finite subcover of Y .


Definition 6.6 ([13]). A space X is said to be weakly F -normal if everypair of disjoint F -closed sets are contained in disjoint open sets.

Theorem 6.7 ([13]). Let f : X → Y be an F -supercontinuous closed surjec-tion. If X is a weakly F -normal space, then Y is a normal space.

7. F -quotient Topology and F -quotient spaces

Let f : X → Y be a surjection from a topological space X onto a set Y . Thequotient topology on Y is the finest topology on Y , which makes f continuous.Several variants of quotient topology have been defined in the literature (see [8,10, 11, 15, 22, and 23]) which in general are weaker than quotient topology andcoincide with the quotient topology if the domain is suitably augmented. Forinterrelations among these variants of quotient topology we refer the interestedreader to [15]. In this section we introduce the notion of F -quotient topologywhich in general lies strictly between the quotient topology and the z-quotienttopology [8].

We may recall that a set U in a space X is said to be z-open if it is expressibleas a union of cozero sets in X .

Definition 7.1. Let p : X → Y be a surjection from a topological space Xonto a set Y .

(i) The collection τ of all subsets A ⊂ Y such that p−1(A) is z-open in X isa topology on Y and is called z-quotient topology [8] and the map p iscalled the z-quotient map.

(ii) The collection τ of all subsets A ⊂ Y such that p−1(A) is F -open in Xis a topology on Y and is called F -quotient topology and the map p iscalled the F -quotient map.

Clearly, z-quotient topology ⊂ F -quotient topology ⊂ quotient topology.

However, none of the above inclusions are reversible as is well exhibited bythe following examples.

Example 7.2. Let (X, τ) be the space of Example 3.8. Then X is a functionallyregular space which is not a completely regular space. Let Y = X and let pdenote the identity map defined on X. Then p is an F -supercontinuous functionwhich is not z-supercontinuous. The F -quotient topology on Y is identical withτ while z-quotient topology is strictly coarser than τ .

Example 7.3. Let X be the space of all positive integers endowed with theprime integer topology σ [24, Example 61, p. 82]. Then X is a Hausdorff spacewhich is not a functionally Hausdorff space and hence not a functionally regularspace. Let Y = X and let p denote the identity map defined on X. Then thequotient topology on Y is same as prime integer topology σ but F -quotienttopology on Y is strictly coarser than σ.

Theorem 7.4. Let p : X → Y be a surjection from a topological space X ontoa topological space (Y, τ), where τ is the F -quotient topology on Y . Then p


is F -supercontinuous. Moreover, τ is the finest topology on Y which makesp : X → Y, F -supercontinuous.

Proof. F -supercontinuity of p is an immediate consequence of the definition ofF -quotient topology. Now let τ1 be a topology on Y such that p : X → (Y, τ1)is F -supercontinuous. Let G be a τ1 open set in Y . By F -supercontinuity ofp, p−1(G) is F -open in X . Now by the definition of F -quotient topology, G isτ -open and hence τ1 ⊂ τ .

In contrast with the quotient space, the following result shows that a functionout of an F -quotient space is continuous if and only if its composition with theF -quotient map is F -supercontinuous.

Theorem 7.5. Let p : X → Y be an F -quotient map. Then a function g :Y → Z is continuous if and only if g p is F -supercontinuous.

Proof. Let U be an open set in Z and g p is F -supercontinuous, then (g p)−1(U) = p−1(g−1(U)), is F -open in X . Since p is an F -quotient map, g−1(U)is open in Y . Hence, g is continuous.

The converse is immediate.

8. Change of Topology

If the topology of domain of an F -supercontinuous function is changed inan appropriate way, then f is simply a continuous function. For, let (X, τ) bea topological space, and let β denote the collection of all F -open subsets of(X, τ). Since the intersection of two F -open sets is F -open, the collection β isa base for a topology τF on X . Indeed β = τF and τF ⊂ τ . The space (X, τ)is functionally regular if and only if τF = τ .

Further, if Bz denotes the collection of all cozero subsets of (X, τ), then Bz

is a base for a topology τz on X . It is immediate that τz ⊂ τF ⊂ τ and thespace X is completely regular if and only if τ = τz . Thus for a completelyregular space τz = τF = τ .

Throughout the section, the symbol τF will have the same meaning as inthe above paragraph.

Remark 8.1. Any topological property which is invariant under continuousbijection will be transferred from (X, τ) to (X, τF ). The list of such propertiesis fairly long. In particular, if (X, τ) is compact Lindelof or countably compact,pseudocompact or quasicompact [5], D-compact or D∗-compact or Dδ-compact[12], separable, connected or pathwise connected, then so is (X, τF ).

Theorem 8.2. A function f : (X, τ) → (Y,ℑ) is F -supercontinuous if andonly if f : (X, τF ) → (Y,ℑ) is continuous.

Many of the results studied in preceding sections follow now from above the-orem and the corresponding standard properties of continuous functions.

Theorem 8.3. Let (X, τ) be a topological space. Then the following statementsare equivalent.


(a) (X, τ) is functionally regular.(b) Every continuous function from (X, τ) into a space (Y,ℑ) is F -supercontinuous.

Proof. (a) ⇒(b) is obvious.(b)⇒(a): Take (Y,ℑ) = (X, τ). Then the identity function 1x on X is contin-uous, and hence F -supercontinuous. Hence by Theorem 8.2, 1x : (X, τF ) →(X, τ) is continuous. Since U ∈ τ implies 1−1

x (U) = U ∈ τF , therefore τ ⊂ τF .Thus it follows that τ = τF , and so (X, τ) is a functionally regular.

Definition 8.4 ([9]). A function f : X → Y from a topological space X intoa topological space Y is said to be F -continuous if for each x ∈ X and eachF -open set U containing f(x) there exists an open set V containing x such thatf(V ) ⊂ U .

Theorem 8.5. Let f : (X, τ) → (Y,ℑ) be a function. Then

(a) f is F -continuous if and only if f : (X, τ) → (Y,ℑF ) is continuous.(b) f is F -open if and only if f : (X, τF ) → (Y,ℑ) is open.

In view of Theorems 8.2 and 8.3, Theorem 4.8 can be restated as follows. Iff : (X, τF ) → (Y,ℑ) is a continuous open surjection and g : (Y,ℑ) → (Z, v) isa function, then g is continuous if and only if gof is continuous.

Moreover, F -quotient topology on Y determined by the surjection f : (X, τ) →Y in Section 7 coincides with usual quotient topology on Y determined byf : (X, τF ) → Y .

References

[1] C. E. Aull, Notes on separation by continuous functions, Indag. Math. 31 (1969), 458–461.

[2] C. E. Aull, Functionally regular spaces, Indag. Math. 38 (1976), 281–288.[3] N. Bourbaki, Elements of General Topology Part I, Hermann, Addison-Wesley, 1966.[4] H. Brandenburg, On spaces with Gδ-basis, Arch. Math. 35 (1980), 544–547.[5] Z. Froli’k Generalization of compact and Lindelof spaces, Czechoslovak. Math. J. 13 (84)

(1959), 172–217 (Russian).[6] N.C. Heldermann, Developability and some new regularity axioms, Can. J. Math. 33,

no. 3 (1981), 641–663.[7] E. Hewitt, On two problems of Urysohn, Ann. of Math. 47, no. 3 (1946), 503–509.[8] J. K. Kohli and R. Kumar, z-supercontinuous functions, Indian J. Pure Appl. Math.

33, no. 7 (2002), 1097–1108.[9] J. K. Kohli, D. Singh, R. Kumar and J. Aggarwal, Between continuity and set connect-

edness, preprint.[10] J. K. Kohli and D. Singh, D-supercontinuous functions, Indian J. Pure Appl. Math. 32,

no. 2 (2001), 227–235.[11] J. K. Kohli and D. Singh, Dδ-supercontinuous functions, Indian J. Pure Appl. Math.

34, no. 7 (2003), 1089–1100.[12] J. K. Kohli and D. Singh, Between compactness and quasicompactness, Acta Math.

Hungar. 106, no. 4 (2005), 317–329.[13] J. K. Kohli, D. Singh and J. Aggarwal, On certain weak variants of normality and

factorizations of normality, preprint.[14] J. K. Kohli, D. Singh and J. Aggarwal, R-supercontinuous functions, communicated.[15] J. K. Kohli, D. Singh and R. Kumar, Some properties of strongly θ-continuous functions,

Bull. Cal. Math. Soc. 100 (2008), 185–196.


[16] N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67 (1960), 269.[17] J. Mack, Countable paracompactness and weak normality properties, Trans. Amer. Math.

Soc. 148 (1970), 265–272.[18] B. M. Munshi and D. S. Bassan, Super-continuous mappings, Indian J. Pure Appl. Math.

13 (1982), 229–236.[19] T. Noiri, On δ-continuous functions, J. Korean Math. Soc. 16 (1980), 161–166.[20] T. Noiri, Supercontinuity and some strong forms of continuity, Indian J. Pure. Appl.

Math. 15, no. 3 (1984), 241–250.[21] I. L. Reilly and M. K. Vamanamurthy, On super-continuous mappings, Indian J. Pure.

Appl. Math. 14, no. 6 (1983), 767–772.[22] D. Singh, D∗-supercontinuous functions, Bull. Cal. Math. Soc. 94, no. 2 (2002), 67–76.[23] D. Singh, cl-supercontinuous functions, Applied General Topology 8, no. 2 (2007), 293–

300.[24] L. A. Steen and J. A. Seeback, Jr., Counter Examples in Topology, Springer Verlag,

New York, 1978.[25] W. T. Van Est and H. Freudenthal, Trennung durch stetige Funktionen in topologischen

Raumen, Indagationes Math. 15 (1951), 359–368.

Received June 2008


J. K. Kohli (jk [email protected])Department of Mathematics, Hindu College, University of Delhi, Delhi 110 007,India.

D. Singh ([email protected])Department of Mathematics, Sri Aurobindo College, University of Delhi-SouthCampus, Delhi 110 017, India.

Jeetendra Aggarwal ([email protected])Department of Mathematics, University of Delhi, Delhi 110 007, India.



Volume 10, No. 1, 2009

pp. 85-119

Arnautov’s problems on semitopologicalisomorphisms

Dikran Dikranjan and Anna Giordano Bruno

Abstract. Semitopological isomorphisms of topological groupswere introduced by Arnautov [2], who posed several questions related tocompositions of semitopological isomorphisms and about the groups G

(we call them Arnautov groups) such that for every group topology τ onG every semitopological isomorphism with domain (G, τ ) is necessarilyopen (i.e., a topological isomorphism). We propose a different approachto these problems by introducing appropriate new notions, necessaryfor a deeper understanding of Arnautov groups. This allows us to findsome partial answers and many examples. In particular, we discuss therelation with minimal groups and non-topologizable groups.

2000 AMS Classification: Primary 22A05, 54H11; Secondary: 18A20, 20F38,20K45.

Keywords: A-complete topology, Heisenberg group, Markov group, minimalgroup, open mapping theorem, permutations group, semitopological isomor-phism, Taımanov topology, topologizable group.

1. Introduction

It is easy to prove that for every continuous isomorphism f : (G, τ) → (H, σ)

of topological groups, there exist a topological group (G, τ ) containing G as a

topological subgroup and an open continuous homomorphism f : (G, τ ) →(H, σ) extending f [2, Theorem 1] (see also [14, Theorem 1.1] for continuoussurjective homomorphisms).

The following notion is motivated by the fact that it is not always possible

to prove the existence of such G and f , asking G to be also a normal subgroup

of G (see also [1] for topological rings).

86 D. Dikranjan and A. Giordano Bruno

Definition 1.1 ([2, Definition 2]). A continuous isomorphism f : (G, τ) →(H, σ) of topological groups is semitopological if there exist a topological group

(G, τ ) containing G as a topological normal subgroup and an open continuous

homomorphism f : (G, τ ) → (H, σ) extending f .

In other words semitopological isomorphisms are restrictions of open contin-uous surjective homomorphisms to normal subgroups. Obviously the class ofsemitopological isomorphisms contains the class of topological isomorphisms.

Arnautov characterized semitopological isomorphisms [2, Theorem 4]. Wegive his characterization in terms of commutators and of thin subsets, as donein [14].

For a neighborhood U of the neutral element eG of a topological group Gcall a subset M of G U -thin if

⋂x−1Ux : x ∈ M is still a neighborhood of

eG (i.e., there exists a neighborhood U1 of eG in G such that xU1x−1 ⊆ U for

every x ∈ M). The subsets M of G that are U -thin for every U are preciselythe thin sets in the sense of Tkachenko [29, 30]. For example compact sets arethin.

Theorem 1.2 ([2, Theorem 4]). Let (G, τ) and (H, σ) be topological groups.Let f : (G, τ) → (H, σ) be a continuous isomorphism. Then f is semitopologicalif and only if for every U ∈ V(G,τ)(eG):

(a) there exists V ∈ V(H,σ)(eH) such that f−1(V ) is U -thin;

(b) for every g ∈ G there exists Vg ∈ V(H,σ)(eH) such that [g, f−1(Vg)] ⊆ U .

In [14] we extended the notion of semitopological isomorphism introducingsemitopological homomorphisms. We defined new properties and consideredparticular cases in order to give internal conditions, similar to those of Theorem1.2, which are sufficient or necessary for a continuous surjective homomorphismto be semitopological. Finally we established various stability properties ofthe class of all semitopological homomorphisms. Many particular cases areconsidered and they turn out to be useful also in this paper as well as otherparticular results; for those we will give references.

In Section 2 we give general properties of semitopological isomorphisms andsee some stability properties of the class Si of all semitopological isomorphisms.In fact it has been proved in [2] that the class Si is stable under taking sub-groups, quotients and products, but not under taking compositions.

The aim of this paper is to discuss and answer the following problems raisedby Arnautov [2]:

Problem A ([2, Problem 14]). Find groups G such that for every group topol-ogy τ on G every semitopological isomorphism f : (G, τ) → (H, σ), where(H, σ) is a topological group, is open.

Arnautov’s problems on semitopological isomorphisms 87

Problem B ([2, Problem 13]) Let K be a class of topological groups. Find(G, τ) ∈ K such that every semitopological isomorphism f : (G, τ) → (H, σ) inK is open.

The third problem concerns compositions:

Problem C ([2, Problem 15])

(a) Which are the continuous isomorphisms of topological groups that arecompositions of semitopological isomorphisms?

(b) Is every continuous isomorphism of topological groups composition ofsemitopological isomorphisms?

1.1. The Open Mapping Theorem and its weaker versions. Accordingto the Banach’s open mapping theorem every surjective continuous linear mapbetween Banach spaces is open [3]. As a generalization, Ptak [22] introducedthe notion of B-completeness for the class of linear topological spaces. It wasbased on the property weaker than openness, that can be formulated also in thelarger class of topological groups as follows: a homomorphism f : G → H oftopological groups is called almost open, if for every neighborhood U of eG in Gthe image f(U) is dense in some neighborhood of eH in H . A topological groupG is B-complete (respectively, Br-complete) if every continuous almost opensurjective homorphism (respectively, isomorphism) from G to any Hausdorffgroup is open. These groups were intensively studied in the sixties and theseventies ([4], [15], [16], [27]). It was shown by Husain [16] that locally compactgroups as well as complete metrizable groups are B-complete. Brown [4] founda common generalization of this fact by proving that Cech-complete groups areB-complete.

The following notion introduced by Choquet (see Doıtchinov [11]) and Ste-phenson [26] in 1970 takes us closer to the spirit of Banach’s open mappingtheorem:

Definition 1.3. A Hausdorff group topology τ on a group G is minimal iffor every continuous isomorphism f : (G, τ) → H, where H is a Hausdorfftopological group, f is a topological isomorphism. Call G totally minimal if forevery continuous homomorphism f : (G, τ) → H, where H is Hausdorff, f isopen.

Clearly, the totally minimal groups are precisely the topological groups thatsatisfy the Banach’s open mapping theorem. Since all surjective homomor-phisms between precompact groups are almost open, a precompact group isBr-complete (respectively, B-complete) if and only if it is minimal (respec-tively, totally minimal). In particular, the Br-complete precompact abeliangroups coincide with the minimal abelian groups as every minimal abeliangroup is precompact according to the celebrated Prodanov-Stoyanov’s theo-rem. According to this theorem, an infinite minimal abelian group is neverdiscrete. This radically changes in the non-abelian case. In the forties Markov


asked whether every infinite group G is topologizable (i.e., admits a non-discreteHausdorff group topology).

Definition 1.4. A group G is:

• Markov if the discrete topology δG is the unique Hausdorff group topol-ogy on G (i.e., δG is minimal);

• totally Markov if G/N is Markov for every N ⊳ G.

Obviously totally Markov implies Markov and finite groups are totally Mar-kov, while every simple Markov group is totally Markov. Denote by M andMt the classes of all Markov and totally Markov groups respectively. Markov’squestion (on whether M contains infinite groups), was answered only thirty-fiveyears later by Shelah [24] (who needed CH for his example, resolving simul-taneously also Kurosh’ problem) and Ol′shanskii [21] (who made use of theproperties of remarkable Adian’s groups).

A smaller class of groups arose in the solution of a specific problem relatedto categorical compactness in [10]: namely the subclass of Mt consisting ofthose groups G ∈ Mt such that every subgroup of G belongs to Mt as well(these groups were named hereditarily non-topologizable by Lukacs [18]). It isstill an open question whether an infinite hereditarily non-topologizable groupexists ([9, 10, 18]).

A possibility to relax the strong requirement in the open mapping theoremin the definition of minimal groups is to restrict the class of topological groups:

Definition 1.5. Let K be a class of topological groups. A topological group(G, τ) ∈ K is K-minimal if (G, σ) ∈ K and σ ≤ τ imply τ = σ.

When K is the class of all metrizable abelian groups, K-minimal groups areprecisely the minimal abelian groups that are metrizable [8], but in general a K-minimal group need not be minimal. Anyway, if H is the class of all Hausdorfftopological groups, then H-minimality is precisely the usual minimality.

Recently new generalizations of minimality for topological groups were con-sidered (relative minimality and co-minimality, cf. [7, 25]).

1.2. Main Results. The next definition reminds the Br-completeness (sincewe impose openness only on certain continuous isomorphisms, namely, the semi-topological ones):

Definition 1.6. A group topology τ on G is A-complete if for every grouptopology σ on G, σ ≤ τ and idG : (G, τ) → (G, σ) semitopological imply τ = σ.

Finally, we can formulate the notion that captures the core of Problem A:

Definition 1.7. A group G is an Arnautov group if every group topology on Gis A-complete (i.e., if for every pair of group topologies τ, σ on G with σ < τ ,idG : (G, τ) → (G, σ) is not semitopological).

Hence Problem A can be formulate also as follows: characterize the groupsG such that every group topology on G is A-complete, that is, characterize theArnautov groups.


We denote by A the class of all Arnautov groups.

Taımanov [28] introduced the group topology TG on a group G, which hasthe family of the centralizers of the elements of G as a prebase of the filter ofthe neighborhoods of eG. This topology was introduced with the aim of thetopologization of abstract groups with Hausdorff group topologies.

Since idG : (G, δG) → (G, σ) is semitopological if and only if σ ≥ TG (see[14, Corollary 5.3] or Remark 5.12) and we are studying Arnautov groups, weneed to impose that TG is discrete and we introduce the following notion.

Definition 1.8. A group G is:

• Taımanov if TG = δG;• totally Taımanov if G/N is Taımanov for every N ⊳ G.

Obviously every simple Taımanov group is totally Taımanov.We denote by T and Tt the classes of Taımanov and totally Taımanov groups

respectively.

Since Problem A in its full generality seems to be hard to handle (because oftwo universal quantifiers), we start considering a particular case, that is whenthe discrete topology on a group G is A-complete and we prove that for a groupG the discrete topology is A-complete if and only if G ∈ T (see Theorem 5.13).Moreover we extend this result for almost trivial topologies (which are obtainedfrom the trivial ones by extension, as their name suggests — see Section 3),characterizing in Theorem 5.15 when an almost trivial topology is A-completein terms of T.

Moreover Tt contains A, but we do not know if they coincide (see Theorem5.16 and Question 5.17).

Example 5.18 considers properties of the permutations group S(Z) relatedto Problem A. First of all it shows that A-completeness has a behavior differentfrom that of the usual minimality. Indeed we see that S(Z) admits at least twodifferent but comparable A-complete group topologies. Moreover S(Z) is notTaımanov and consequently not Arnautov. Nevertheless S(Z)/Sω(Z) is totallyTaımanov but we do not know if it is also Arnautov (see Question 5.20).

This question can be seen as a first step in answering the following one,which could give an infinite example of a simple infinite Markov group withoutassuming CH (see Question 5.27):

does S(Z)/Sω(Z) ∈ M?

But the situation can be reversed: if S(Z)/Sω(Z) ∈ M then S(Z)/Sω(Z) ∈ A,in view of Corollary 5.26(b), which says that every simple Markov group isnecessarily Arnautov. Thanks to this property we have the unique infiniteArnautov group that we know at the moment, that is Shelah group, which isan infinite simple Markov group constructed under CH [24] (see Example 5.29).

The next definition, combining Definition 1.6 (A-completeness) and Defini-tion 1.5 (K-minimality) will allow us to handle easier Problem B.


Definition 1.9. For a class K of topological groups, a topological group (G, τ)from K is AK-complete if (G, σ) ∈ K, σ ≤ τ and idG : (G, τ) → (G, σ)semitopological imply τ = σ.

Let G be the class of all topological groups.

Remark 1.10.

(a) Obviously K-minimality implies AK-completeness and K-minimalitycoincides with AK-completeness whenever all groups in K ⊆ G areabelian.

(b) Moreover A-completeness coincides with AG-completeness. So ProblemA can be seen as a particular case of Problem B, namely with K = G.

(c) If K ⊆ K′ are classes of topological groups, then for every G ∈ K AK′ -complete implies AK-complete. In particular, if K ⊆ G and G ∈ K,then G A-complete implies G AK-complete.

Clearly AH-completeness is a generalization of minimality, since H-minima-lity is precisely the usual minimality, which is intensively studied, as noted inSection 1.1. This is a strict generalization as shown by Example 6.1.

A topological group G has small invariant neighborhoods (i.e., G is SIN ) ifG is thin (i.e., it has a local base at eG of neighborhoods invariant under con-jugation). We prove that a topological group, which is SIN and AH-complete,is A-complete if and only it has trivial center (see Remark 6.8). In particular,if G is a group with trivial center, its discrete topology is AH-complete if andonly if G ∈ T (see Corollary 6.6). So also in this case Taımanov groups play acentral role.

Moreover we give an example of a small class K in which each element isAK-complete (see Example 6.14). This class is built on the Heisenberg group

HR :=

1 R R

0 1 R

0 0 1

,

that is the group of upper unitriangular 3 × 3 matrices over R, endowed withdifferent group topologies. The group HR is nilpotent of class 2.

In a forthcoming paper [6] we extend this example for generalized Heiseberggroups, that is, the group of upper unitriangular 3× 3 matrices over a unitaryring A.

In Example 7.5 we resolve negatively item (b) of Problem C. Moreover The-orem 7.2 answers partially (a), in the case when the topologies on the domainand on the codomain are the discrete and the indiscrete one respectively. Sincewe consider the trivial topologies, the condition that we find is exclusively al-gebraic. Indeed we prove that idG : (G, δG) → (G, ιG) is composition of nsemitopological isomorphisms if and only if G is nilpotent of class ≤ n, wheren ∈ N+.


Notation and terminology. We denote by R, Q, Z, P, N and N+ respectivelythe field of real numbers, the field of rational numbers, the ring of integers, theset of primes, the set of natural numbers and the set of positive integers.

Let G be a group and x, y ∈ G. We denote by [x, y] the commutator of xand y in G, that is [x, y] = xyx−1y−1 and for x ∈ G and a subset Y of G let[x, Y ] = [x, y] : y ∈ Y . More in general, if H and K are subgroups of G, let

[H, K] = 〈[h, k] : h ∈ H, k ∈ K〉,

and in particular the derived G′ of G is G′ = [G, G], that is, the subgroup ofG generated by all commutators of elements of G. The center of G is Z(G) =x ∈ G : xg = gx, ∀g ∈ G and for g ∈ G the centralizer of g in G is cG(g) =x ∈ G : xg = gx.

The diagonal map ∆ : G → G×G is defined by ∆(g) = (g, g) for every g ∈ G.If H is another group, we denote by p1 : G × H → G and p2 : G × H → Hthe canonical projections on the first and the second component respectively.If f : G → H is a homomorphism, denote by Γf the graph of f , that is thesubgroup Γf = (g, f(g)) : g ∈ G of G × H .

If τ is a group topology on G then denote by V(G,τ)(eG) the filter of allneighborhoods of eG in (G, τ) and by Bτ a base of V(G,τ)(eG). If X is a subset

of G, Xτ

stands for the closure of X in (G, τ).If N is a normal subgroup of G and π : G → G/N is the canonical projec-

tion, then τq is the quotient topology of τ in G/N . Moreover Nτ denotes the

subgroup eGτ. The discrete topology on G is δG and the indiscrete topology

on G is ιG.For undefined terms see [12, 13].

2. Properties of semitopological isomorphisms

In the next remark we discuss the possibility to consider only the case of onegroup G endowed with two different topologies τ ≥ σ taking idG : (G, τ) →(G, σ) as the continuous isomorphism:

Remark 2.1. Let (G, τ), (H, η) be topological groups and f : (G, τ) → (H, η)a continuous isomorphism. Consider the topology σ = f−1(η) on G. Thistopology σ is coarser than τ and so idG : (G, τ) → (G, σ) is a continuousisomorphism and (G, σ) is topologically isomorphic to (H, η). In particular

idG : (G, τ) → (G, σ) is semitopological if and only iff : (G, τ) → (H, η) is semitopological.

Moreover the next proposition shows that semitopological is a “local” prop-erty, like the stronger property open. The proof is a simple application ofTheorem 1.2.

Proposition 2.2. Let G be a group and τ, σ group topologies on G such thatσ ≤ τ . Then idG : (G, τ) → (G, σ) is semitopological if there exists a τ-opensubgroup N of G such that idG N : (N, τ N ) → (N, σ N ) is semitopological.


The following theorems show the stability of the class of semitopologicalisomorphisms under taking subgroups, quotients and products.

Theorem 2.3 ([2, Theorems 7 and 8]). Let G be a group, σ ≤ τ group topolo-gies on G and suppose that idG : (G, τ) → (G, σ) is semitopological.

(a) If A is a subgroup of G, then idA : (A, τ A) → (A, σ A) is semitopo-logical.

(b) If A is a normal subgroup of G, then idG/A : (G/A, τq) → (G/A, σq) issemitopological.

Theorem 2.4 ([2, Theorem 9], [14, Theorem 6.15]). Let Gi : i ∈ I be afamily of groups and τi : i ∈ I, σi : i ∈ I families of group topologiessuch that σi ≤ τi are group topologies on Gi for every i ∈ I. Then idGi :(Gi, τi) → (Gi, σi) is semitopological for every i ∈ I if and only if

∏i∈I idGi :∏

i∈I(Gi, τi) →∏

i∈I(Gi, σi) is semitopological.

The next lemma shows a cancellability property of compositions of semi-topological isomorphisms.

Lemma 2.5 ([14, Theorem 6.11]). Let σ ≤ τ be group topologies on a groupG. If idG : (G, τ) → (G, σ) is semitopological, then for a group topology ρ onG such that σ ≤ ρ ≤ τ , idG : (G, τ) → (G, ρ) is semitopological.

In a particular case, that is for initial topologies, the converse implication ofTheorem 2.3(b) holds true:

Lemma 2.6. Let G be a group and N a normal subgroup of G. Let σ ≤ τbe group topologies on G/N and σi ≤ τi the respective initial topologies onG. Then idG : (G, τi) → (G, σi) is semitopological if and only if idG/N :(G/N, τ) → (G/N, σ) is semitopological.

In the next theorem we consider the particular cases when one of the twotopologies on G is trivial:

Theorem 2.7 ([2, Corollary 5], [14, Corollary 5.11]). Let G be a group and τa group topology on G. Then:

(a) idG : (G, δG) → (G, τ) is semitopological if and only if cG(g) is τ-openfor every g ∈ G;

(b) idG : (G, τ) → (G, ιG) is semitopological if and only if G′ ≤ Nτ .

Since Z(G) ⊆ cG(g) for every g ∈ G, by (a) idG : (G, δG) → (G, τ) semi-topological implies Z(G) τ -open.

The condition G′ ≤ Nτ in (b) is equivalent to say that G′ is indiscreteendowed with the topology inherited from (G, τ). Moreover, as noted in [14],it implies that (G, τ) is SIN.

For SIN groups condition (a) of Theorem 1.2 is always verified, since SINgroups are thin, so only condition (b) remains:


Proposition 2.8. Let G be a group and σ ≤ τ group topologies on G. Supposethat (G, τ) is SIN. Then idG : (G, τ) → (G, σ) is semitopological if and only iffor every U ∈ V(G,τ)(eG) and for every g ∈ G there exists Vg ∈ V(G,σ)(eG) suchthat [g, Vg] ⊆ U .

The next lemma gives a simple necessary condition of algebraic nature for acontinuous isomorphism to be semitopological.

Lemma 2.9. Let G be a group and σ ≤ τ group topologies on G, such thatidG : (G, τ) → (G, σ) is semitopological. Then [G, Nσ] ≤ Nτ .

Proof. By Theorem 1.2, for every U ∈ V(G,τ)(eG) and every g ∈ G, there existsVg ∈ V(G,τ)(eG) such that [g, Vg] ⊆ U . Consequently [g, Nσ] ⊆ U for everyg ∈ G, so [g, Nσ] ⊆ Nτ for every g ∈ G and hence [G, Nσ] ≤ Nτ .

Corollary 2.10. Let G be a group and τ a group topology on G. If τ isHausdorff, then idG : (G, τ) → (G, ιG) is semitopological if and only if G isabelian.

Proof. If idG : (G, τ) → (G, ιG) is semitopological, by Lemma 2.9 G′ ≤ Nτ =eG and hence G is abelian. If G is abelian every continuous isomorphism issemitopological.

In particular idG : (G, δG) → (G, ιG) is semitopological if and only if thegroup G is abelian.

Proposition 2.11. Let G be a group and σ ≤ τ group topologies on G, suchthat idG : (G, τ) → (H, σ) is semitopological. If Z(G) = eG and τ is Haus-dorff, then σ is Hausdorff as well.

Proof. Since Nτ = eG and [G, Nσ] ≤ Nτ by Lemma 2.9, using the hypothesisZ(G) = eG we conclude that Nσ = eG.

3. Almost trivial topologies

In this section we introduce a class of group topologies containing the trivialones and with nice stability properties; moreover we extend Theorem 2.7 tothis class.

Definition 3.1. [14, Definition 5.13] A topological group (G, τ) is almost trivialif Nτ is open in (G, τ).

Since in this case τ is completely determined by the normal subgroup N :=Nτ of G, we denote an almost trivial topology on G by ζN , underling the roleof the normal subgroup.

Every group topology on a finite group is almost trivial and every almosttrivial group is SIN.

For example, for a group G, the discrete and the indiscrete topologies (i.e.,the so-called trivial topologies) are almost trivial, with δG = ζeG and ιG = ζG.This justifies the term used in Definition 3.1.


Lemma 3.2. Let G be a simple non-abelian group and let τ be a group topologyon G. Then either Nτ = G or Nτ = eG, that is, either τ = ιG or τis Hausdorff, respectively. If τ is almost trivial, then τ is either discrete orindiscrete.

The almost trivial topologies help also to express in simple terms topologicalproperties:

Remark 3.3. Given a topological group (G, τ) and a normal subgroup N ofG, it is possible to consider the group topology obtained “adding” to the openneighborhoods also N (since it is normal, it suffices to add N to the prebase ofthe neighborhoods and all the intersections U ∩ N , with U ∈ V(G,τ)(eG), givethe neighborhoods of eG in the new topology). This new topology is supτ, ζN.

For example, if G is a group and τ its profinite topology, with Bτ = Nαα,where the Nα are all the normal subgroups of G of finite index, then τ =supα ζNα . More in general, if τ is a linear topology on G, that is Bτ = Nαα,where Nα are normal subgroups of G, then τ = supα ζNα .

If (G, τ) is a topological group, let τ denote the quotient topology of (G, τ)with respect to the normal subgroup Nτ , which is indiscrete. Then τ is Haus-dorff. Moreover (G, τ) is almost trivial if and only if (G/Nτ , τ ) is discrete.

Analogously it is possible to consider the case when a topological group(G, τ) has a discrete normal subgroup D such that (G/D, τq) is indiscrete. Forgroups with this property we have a strong consequence:

Lemma 3.4. Let (G, τ) be a topological group such that D is a discrete normalsubgroup of (G, τ) and (G/D, τq) is indiscrete. Then (G, τ) ∼= D × Nτ , whereD is discrete and Nτ is indiscrete. In particular τ is almost trivial.

Proof. Pick a symmetric neighborhood W of eG in G such that W 3∩D = eG.Since (G/D, τq) is indiscrete, D is dense in G, so G = DW . Let w1, w2 ∈ W .Then there exists d ∈ D such that w1w2 ∈ dW . Let w1w2 = dw for somew ∈ W . Then d = w1w2w

−1 ∈ W 3 ∩ D = eG. So w1w2 = w ∈ W .Since W is symmetric, this proves that W is an open subgroup of M withW ∩ D = eG. Hence the restriction of the canonical projection G → G/Dto W gives a topological isomorphism W ∼= (G/D, τq). This shows that W isan indiscrete group. Since Nτ ≤ W is closed, we deduce that W = Nτ . Thisproves that Nτ is open in τ and that (G, τ) ∼= D × Nτ .

3.1. Permanence properties of the almost trivial topologies. The as-signment N 7→ ζN defines an order reversing bijection between the completelattice N (G) of all normal subgroups of a group G and the complete latticeAT (G) of all almost trivial group topologies on G. Let us note that the com-plete lattice AT (G) is not a sublattice of the complete lattice T (G) of all grouptopologies on G. Indeed, the meet of a family ζNi : i ∈ I in AT (G) is simplyζ⋂

i∈I Ni, whereas the meet of a family ζNi : i ∈ I in T (G) is the group topol-

ogy having as prebase of the neighborhoods at eG the family ζNi : i ∈ I (in


other words, the latter topology may be strictly weaker than the former one incase I is infinite).

The next lemma shows, among others, that the class of almost trivial groupsis closed under taking subgroups and quotients.

Lemma 3.5. Let (G, ζN ) be an almost trivial group, where N is a normalsubgroup of G.

(a) For every subgroup H of G:(a1) the topology induced on H by ζN is almost trivial and coincides

with ζH∩N ;(a2) the following conditions are equivalent: (i) H is ζN -open; (ii) H

is ζN -closed; (iii) H ≥ N .(b) For every normal subgroup N0 of G the quotient topology of ζN on

G/N0 is almost trivial and coincides with ζN0N/N0.

Remark 3.6. In connection to item (a1) of the previous lemma notice that ifH is an open subgroup of a topological group G and H is almost trivial, thenalso G is almost trivial.

Now we show that the class of almost trivial groups is stable also with respectto taking finite products.

Lemma 3.7. Let G1, G2 be groups and N1, N2 normal subgroups of G1, G2

respectively. Then ζN1× ζN2

= ζN1×N2on G1 × G2.

The next lemma follows directly from the definitions.

Lemma 3.8. Let G be a topological group and N an indiscrete normal subgroupof G such that G/N is almost trivial. Then G is almost trivial.

We want to generalize this lemma and we need the following concept.

Definition 3.9. For a class of topological groups P one says that P has thethree space property, if a topological group G belongs to P whenever N ∈ Pand G/N ∈ P for some normal subgroup N of G.

For example the class of all discrete groups and the class of all indiscretegroups have the three space property. So the next result shows that the classof all almost trivial groups is the smaller class with the three space propertycontaining all discrete and all indiscrete groups.

Proposition 3.10. The class of almost trivial groups has the three space prop-erty.

Proof. We have to prove that, in case G is a group and N a normal subgroup ofG, if τ is a group topology on G such that (N, τ N) and (G/N, τq) are almosttrivial, then (G, τ) is almost trivial.

Let M be the normal subgroup of G containing N such that M/N = Nτq .Then M/N is indiscrete and open in G/N . Consequently, M is open in (G, τ).To end the proof we need to verify that M is almost trivial (see Remark 3.6).


If τ N is Hausdorff, equivalently it is discrete, since it is almost trivial,and by Lemma 3.4 M is almost trivial. So we consider now the general case.The subgroup N1 := Nτ ∩ N is the closure of eG in M . Then N1 is anormal subgroup of N . Now the normal subgroup N/N1 of the Hausdorffquotient group M/N1 is almost trivial and consequently discrete. Moreover,the quotient (M/N1)/(N/N1) ∼= M/N is indiscrete. So by the previous casethe group M/N1 is almost trivial. Since the group N1 is indiscrete, we canconclude with Lemma 3.8.

3.2. Semitopological isomorphisms between almost trivial topologies.

Since every almost trivial group is SIN, it is possible to apply Proposition 2.8instead of Theorem 1.2 to verify if a continuous isomorphism is semitopological.In case the topology on the domain or that on the codomain is almost trivial,the conditions of Theorem 1.2 become simpler:

Proposition 3.11. Let (G, σ) be a topological group and let σ ≤ τ be grouptopologies on G.

(a) If τ is almost trivial, then idG : (G, τ) → (G, σ) is semitopological ifand only if for every g ∈ G there exists Vg ∈ V(G,σ)(eG) such that[g, Vg] ⊆ Nτ .

(b) If σ is almost trivial, then idG : (G, τ) → (G, σ) is semitopological ifand only if Nσ is U -thin for every U ∈ V(G,τ)(eG) and [G, Nσ] ≤ Nτ .

Proof. (a) follows from Proposition 2.8.(b) The necessity of the condition that Nσ is U -thin for every U ∈ V(G,τ)(eG)

follows from Theorem 1.2, while the necessity of [G, Nσ] ≤ Nτ follows fromLemma 2.9. The sufficiency of the two conditions is a consequence of Theorem1.2.

If τ is Hausdorff in this proposition, then (b) becomes N ≤ Z(G). So we havethe following corollary, which can be also seen as a consequence of Proposition2.11.

Corollary 3.12. Let G be a group. If τ is a Hausdorff group topology on G,then for every non-central τ-open subgroup N of G idG : (G, τ) → (G, ζN ) isnot semitopological.

Combining together the two items of Proposition 3.11 we have preciselythe following corollary, which is the “almost trivial version” of Theorem 1.2.Furthermore it shows that the necessary condition of Lemma 2.9 becomes alsosufficient in the case of almost trivial topologies.

Corollary 3.13. [14, Lemma 5.15] Let G be a group and ζN ≥ ζL almosttrivial group topologies on G. Then idG : (G, ζN ) → (G, ζL) is semitopologicalif and only if [G, L] ≤ N .

The next example is a consequence of this corollary.


Example 3.14. Let G be a group and ζN an almost trivial group topology onG. Consider

(G, δG)idG−−→ (G, ζN )

idG−−→ (G, ιG).

Then:

(a) idG : (G, δG) → (G, ζN ) is semitopological if and only if N ≤ Z(G);(b) idG : (G, ζN ) → (G, ιG) is semitopological if and only if G′ ≤ N .

On a group G it is possible to consider the almost trivial topology generatedby G′, that is ζG′ . A group G is perfect if G = G′, and G is perfect if and onlyif ζG′ = ιG.

Remark 3.15. With this topology generated by the derived group, we canwrite again Theorem 2.7(b) as:

Let G be a group and τ a group topology on G. Then idG :(G, τ) → (G, ιG) is semitopological if and only if τ ≤ ζG′ .

Remark 3.16. Let N be a normal subgroup of a group G and let ζN be therespective almost trivial topology on G.

Let τ be a group topology on G. Then idG : (G, ζNτ ) → (G, τ) is con-tinuous. Moreover, if ζL is another almost trivial topology on G such thatidG : (G, ζL) → (G, τ) is continuous, then idG : (G, ζL) → (G, ζNτ ) is continu-ous.

(G, ζNτ ) // (G, τ)

(G, ζL)

::uuuuuuuuu

eeKK

KK

K

Consequently idG : (G, ζL) → (G, τ) semitopological implies idG : (G, ζL) →(G, ζNτ ) semitopological by Lemma 2.5, that is, [G, Nτ ] ≤ L by Corollary 3.13.

4. Taımanov groups

Let F ∈ [G]<ω be a finite subset of G and

cG(F ) =⋂

x∈F

cG(x)

the centralizer of F in G. Then C = cG(F ) : F ∈ [G]<ω is a family ofsubgroups of G closed under finite intersections. Then the Taımanov topologyTG has C as local base at eG, that is BTG = C.

We collect in the next lemma the first properties of this topology.

Lemma 4.1. Let G be a group. Then:

(a) NTG = Z(G);(b) TG is Hausdorff if and only if Z(G) = eG;(c) G is abelian if and only if TG = ιG;(d) in case G is finitely generated, TG is almost trivial; in particular TG =

δG if and only if Z(G) is trivial.


4.1. Permanence properties of the class T. The following results showthat the Taımanov topology has nice properties. The next proposition provesthat it is a functorial topology with respect to continuous surjective homomor-phisms.

Proposition 4.2. Let G be a group. Then every surjective homomorphismf : (G, TG) → (H, TH) is continuous.

Proof. Let h ∈ H and consider g ∈ G such that f(g) = h. Then f(cG(g)) ⊆cH(h). This proves the continuity of f : (G, TG) → (H, TH).

On the other hand, the next example shows that the Taımanov topology isnot functorial with respect to open surjective homomorphisms.

Example 4.3. For

HZ :=

1 Z Z

0 1 Z

0 0 1

the group of upper unitriangular 3×3 matrices over Z, the canonical projectionπ : (HZ, THZ

) → (HZ/Z(HZ), THZ/Z(HZ)) is not open.Indeed, since HZ/Z(HZ) =: G is abelian, TG = ιG by Lemma 4.1(c). More-

over note that G ∼= Z × Z. Let h =

1 0 00 1 10 0 1

∈ HZ. Then cHZ

(h) =

1 0 Z

0 1 Z

0 0 1

and π(cHZ

(h)) ∼= 0 × Z, which is not open in (G, ιG).

Lemma 4.4. Let G =∏

i∈I Gi. Then∏

i∈I TGi ≤ TG. If I = 1, . . . , n isfinite, then TG1

× . . . × TGn = TG.

Proof. Since all the canonical projections πi : (G, TG) → (Gi, TGi) are contin-uous by Proposition 4.2,

∏i∈I TGi ≤ TG.

Suppose now that I = 1, . . . , n is finite. If F is a finite subset of G1 ×. . .×Gn, then it is contained in a finite subset of the form F1 × . . .×Fn, whereeach Fi is a finite subset of Gi for i = 1, . . . , n. Moreover cG(F1 × . . . × Fn) =cG1

(F1) × . . . × cGn(Fn). This proves that TG = TG1× . . . × TGn .

Proposition 4.5. (a) If G ∈ T, then Z(G) = eG.(b) If G ∈ Tt, then G is perfect.

Proof. (a) Follows from Lemma 4.1(b).(b) Since G/G′ is abelian and in T, G/G′ is trivial in view of Lemma 4.1(c),

that is G = G′.

It follows from (a) that every non-trivial abelian group G 6∈ T.

The next result about products is a consequence of Lemma 4.4.

Proposition 4.6. The class T is closed under taking finite products.


Proof. Let G1, G2 ∈ T and G := G1 ×G2. By Lemma 4.4 TG = TG1×TG2

andso TG = δG, that is G ∈ T. This can be extended to all finite products.

The next example in particular shows that T is not closed under takingquotients and subgroups since the groups in (b) and (c) have abelian quotients(so they are not in Tt) and non-trivial abelian subgroups.

Example 4.7. (a) A finite group G ∈ T if and only if Z(G) = eG. Thisfollows from Lemma 4.1, but can be also simply directly proved.

(b) Let G =

(R∗ R

0 1

). Then G ∈ T. Indeed, for F =

(2 00 1

),

(1 10 1

)

cG(F ) = eG.(c) Every non-abelian free group F (X) of rank > 1 is in T. Indeed, for F =

a, b, where a, b ∈ X are generators of F (X), cF (X)(F ) = eF (X).

This example shows also that the condition “surjective” in Proposition 4.2cannot be removed: if G is one of the groups in (b) or (c), then G has somenon-trivial abelian subgroup H . Since H is abelian, TH = ιH , while TG = δG.Consequently the injective homomorphism (H, TH) → (G, TG) is far from beingcontinuous.

Remark 4.8. Since non-abelian free groups of rank > 1 are Taımanov (asshown in Example 4.7(c)),

• there exist arbitrarily large Taımanov groups; moreover, every non-abelian subgroup of a non-abelian free group is Taımanov, being free[23];

• every group is quotient of a Taımanov group, since every group is quo-tient of a non-abelian free group of rank > 1 [23].

It is not clear if this holds also for subgroups:

Question 4.9. Is every group subgroup of a Taımanov group?

Theorem 4.11 answers positively the question in the abelian case.For an abelian group G and p ∈ P in what follows we denote by rp(G) the

p-rank of G.

Lemma 4.10. Let G be an abelian group with r2(G) = 0. Then there existsH ∈ T such that G ≤ H and [H : G] = 2.

Proof. Let f : G → G be defined by f(x) = −x for every x ∈ G. Moreover let

H := G ⋊ 〈f〉

(⋊ denotes the semidirect product). Then cH(0, f) = 〈(0, f)〉 and (0, f) 6∈cH(g, idG) for every g ∈ G \ 0. Consequently for F = (g, idG), (0, f), withg ∈ G \ 0, cH(F ) = eH, that is H ∈ T. Since f has order 2, G has index 2in G.

Theorem 4.11. For every abelian group G there exists a group H ∈ T con-taining G as a subgroup and such that |H | = ω · |G|.


Proof. Let G be an abelian group. Then G ⊆ D(G) = G1⊕G2, where r2(G1) =0 and r2(G2) = r2(D(G)). Then there exists H1 ∈ T such that G1 ≤ H1 and|H1| = 2 · |G1| by Lemma 4.10.

Now consider G2 =⊕

r2(G) Z(2∞). If r2(G) ≤ ω, then G2 is contained

in⊕

ω Z(2∞). Let then σ be the shift⊕

ω Z(2∞) →⊕

ω Z(2∞) defined by(xn)n 7→ (xn−1)n for every (xn)n ∈

⊕ω Z(2∞). Then σn has no non-zero fixed

point for every n ∈ Z, n 6= 0.

Claim. Let G be an abelian group and let f be an automorphism of G suchthat fn has no non-zero fixed point for every n ∈ Z, n 6= 0. Then there existsH ∈ T such that G ≤ H and |H | = ω · |G|.

Proof of the claim. Let

H := G ⋊ 〈f〉.

Then cH(0, f) = 〈(0, f)〉 and (0, f) 6∈ cH(g, idG) for every g ∈ G \ 0. Con-sequently for F = (g, idG), (0, f), with g ∈ G \ 0, cH(F ) = eH, that isH ∈ T. Since f has infinite order |H | = ω · |G|.

By the claim there exists a group H2 ∈ T such that G2 ≤⊕

ω Z(2∞) ≤ H2

and |H2| = ω. Suppose that r2(G) ≥ ω. Then G2∼=

⊕r2(G)(

⊕ω Z(2∞)). Let

σ :⊕

r2(G)(⊕

ω Z(2∞)) →⊕

r2(G)(⊕

ω Z(2∞)) be defined by σ ⊕ω Z(2∞)= σ.

Then σn has no non-zero fixed point for every n ∈ Z, n 6= 0, and again theclaim gives a group H2 ∈ T that contains G2 as a subgroup and such that|H2| = |G2|.

Let H := H1 ⊕ H2. By Proposition 4.6 H ∈ T. Moreover |H | = ω · |H1| ·|H2| = ω · |G1| · |G2| = ω · |G|.

Lemma 4.13 shows that to prove that a group is Taımanov it suffices toconsider a convenient quotient with a finite normal subgroup and check whetherit is Taımanov.

Claim 4.12. Let G be a group with Z(G) = eG. If there exists a finite subsetF of G such that cG(F ) is finite, then there exists another finite subset F1 ⊇ Fof G such that cG(F1) = eG. In particular G ∈ T.

Proof. Let cG(F ) = eG, g1, . . . , gn. Since Z(G) = eG, for every i ∈1, . . . , n there exists hi ∈ G such that [gi, hi] 6= eG; in particular gi 6∈ cG(hi).Let F1 = F ∪ h1, . . . , hn. Then gi 6∈ cG(F1) for every i ∈ 1, . . . , n. SincecG(F1) ⊆ cG(F ) = eG, g1, . . . , gn, this proves that cG(F1) = eG.

Lemma 4.13. Let G be a group with Z(G) = eG and let N be a normalfinite subgroup of G such that G/N ∈ T. Then G ∈ T.

Proof. Let F1 be a finite subset of G/N such that cG/N (F1) = eG/N. Letπ : G → G/N be the canonical projection and let F be a finite subset of G suchthat π(F ) = F1. Since π(cG(F )) ⊆ cG/N (F1) = eG/N, cG(F ) ⊆ N . Since Nis finite, Claim 4.12 applies to conclude that G ∈ T.


The next is an example of a totally Taımanov group.

Example 4.14. We denote by G := SO3(R) the group of all orthogonal ma-trices 3 × 3 with determinant 1 and coefficients in R. Then G ∈ T. Since G issimple, G ∈ Tt.

Indeed, G =⋃

α Tα, where Tα∼= T and Tα is generated by an element α of

G, that is, 〈α〉 = Tα. Moreover cG(α) contains Tα as a finite index subgroupand for α, β ∈ G with α 6= β and Tα 6= Tβ, Tα∩Tβ is finite. Then cG(α)∩cG(β)is finite. By Claim 4.12 G ∈ T.

4.2. The permutations groups. For a set X , x ∈ X and a subgroup H ofS(X) let

OH(x) := h(x) : h ∈ H,

Stabx := ρ ∈ S(X) : ρ(x) = x, and

Sx := Stabx ∩ H.

Moreover H induces a partition of X , that is X =⋃

x∈RHOH(x), where RH ⊆

X is a set of representing elements.If τ ∈ S(X), then

Stabx = (Stab τ(x))τ .

Remark 4.15. Let X be a set and H a subgroup of S(X). If τ ∈ NS(X)(H),then:

(a) τ(OH (x)) = OH(τ(x));(b) Sx = (Sτ(x))

τ ; indeed, Sx = Stabx ∩ H = (Stab τ(x))τ ∩ Hτ =(Stab τ(x) ∩ H)τ = (Sτ(x))

τ ;(c) τ induces a permutation τ of RH . Indeed, τ(OH(x)) = OH(τ(x)) by

(a); so we can define τ(x) = y, where y ∈ RH is the representingelement of OH(τ(x)). Then

cS(X)(H) =

τ ∈⋂

x∈RH\supp τ

NH(Sx) · Stabx : [H, τ ] ⊆⋂

x∈supp τ

Sx

.

We describe the subgroups of S(X) with trivial centralizer:

Lemma 4.16. Let X be a set and H a subgroup of S(X). Then cS(X)(H) =idX if and only if the following conditions hold:

(a) Sx = NH(Sx) for every x ∈ RH , and(b) Sx and Sy are not conjugated in H for every x, y ∈ RH with x 6= y.

Proof. Let τ ∈ cS(X)(H)\idX. There exists x ∈ RH such that y := τ(x) 6= x.Indeed, if τ(x) = x for every x ∈ RH , then for every z ∈ X , there exist h ∈ Hand x ∈ RH such that z = h(x), and so τ(z) = τ(h(x)) = h(τ(x)) = h(x) = z.By Remark 4.15(a,b)

τ(OH(x)) = OH(τ(x)) and Sx = (Sy)τ = Sy.


If y ∈ OH(x), then τ OH(x): OH(x) → OH(x) is a bijection and y = h0(x)for some h0 ∈ H ; then τ(h(x)) = h(τ(x)) = hh0(x) for every h ∈ H . Leth ∈ Sx. Since τ is well-defined, h(x) = x implies hh0(x) = h0(x), that is(h0)

−1hh0(x) = x. This is equivalent to hh0 ∈ Sx, that is h0 ∈ NH(Sx). Buth0 6∈ Sx and this contradicts (a).

Suppose now that y 6∈ OH(x) and so OH(x) ∩ OH(y) = ∅. Let z ∈ RH ∩OH(y). Then y = h0(z) for some h0 ∈ H . By Remark 4.15(b) Sz = (Sy)h0 =(Sx)h0 and this contradicts (b).

Assume that there exists h0 ∈ NH(Sx)\Sx for some x ∈ RH . Let τ : X → Xbe defined by τ(x) = h0(x), τ(h(x)) = hh0(x) for every h ∈ H and τ(y) = yfor every y ∈ X \ OH(x). This τ is well-defined. Indeed, if h1(x) = h2(x) forsome h1, h2 ∈ H , that is, h−1

2 h1 ∈ Sx; then h1h0(x) = h2h0(x), equivalently

h−10 (h−1

2 h1)h0(x) = x, that is, h−10 (h−1

2 h1)h0 ∈ Sx, which holds true by thehypothesis that h0 ∈ NH(Sx). Moreover, it is possible to check that τ ∈ S(X).By the definition τh = hτ for every h ∈ H and so idX 6= τ ∈ cS(X)(H).

Suppose that Sx = (Sz)h0 for some x, z ∈ RH and h0 ∈ H . Then for

y = h−10 (z) ∈ OH(z) we have Sy = (Sz)

h0 = Sx by Remark 4.15(b). Defineτ : X → X as τ(x) = y, τ(h(x)) = h(y) for every h ∈ H and τ(w) = w forevery w ∈ X \OH(x). Then τ is well-defined; indeed, if h1(x) = h2(x) for someh1, h2 ∈ H , that is, h−1

2 h1 ∈ Sx, then h1(y) = h2(y), equivalently, h−12 h1 ∈ Sy,

which holds true since Sx = Sy. Moreover, it is possible to check that τ ∈ S(X).By the definition τh = hτ for every h ∈ H and so idX 6= τ ∈ cS(X)(H).

Proposition 4.17. For a cardinal κ the following conditions are equivalent:

(a) there exists a set X with |X | = κ and S(X) ∈ T;(b) there exists a set X with |X | = κ such that there exists a finitely gen-

erated subgroup H of S(X) such that Sx = NH(Sx) for every x ∈ RH

and Sx, Sy are not conjugated for every x, y ∈ RH with x 6= y.

If κ > ω, then the following condition is equivalent to the previous:

(c) there exists a finitely generated group H admitting a family S = Sα :α < κ of subgroups of H such that Sα = NH(Sα) for every α < κ.

Proof. (a)⇔(b) The condition S(X) ∈ T is equivalent to the existence of afinite subset F of S(X) such that cS(X)(F ) = idX. Let H = 〈F 〉. Then

cS(X)(H) = cS(X)(F )

and so equivalently cS(X)(H) = idX. By Lemma 4.16 we have the conclusion.(b)⇒(c) Since κ > ω, and each OH(x) is countable, |RH | = κ. So Sx : x ∈

RH is the family requested in (c).(c)⇒(b) Since κ > ω and H is countable, we can suppose that S has the

property that Sα and Sβ are not conjugated in H for every α, β < κ withα 6= β. Indeed, every subgroup Sα of H has at most countably many conjugatedsubgroups in H , so we can restrict the family S taking only one element forevery class of conjugation, finding a subfamily of the same cardinality κ as S.


Define Xα := hSα : h ∈ H for every α < κ and X :=⋃

α<κ Xα. Moreoverlet xα := idHSα ∈ Xα for every α > κ. In particular |X | = κ. Moreover Hacts on X by multiplication on the left and OH(xα) = Xα for every α < κ.

There exists a group homomorphism ϕ : H → S(X); let H := ϕ(H) ≤ S(X).

Then H is finitely generated and the action of H on X is the same as the actionof H on X . Then OH(xα) = Xα for every α < κ and RH = xα : α < κ.

Moreover ϕ(Sα) = Stabxα ∩ H =: Sxα . Since Sα = NH(Sα) for every α < κand Sα and Sβ are not conjugated for every α < β < κ, it is possible to provethat Sxα = NH(Sxα) for every xα ∈ RH and Sxα and Sxβ

are not conjugatedfor every xα, xβ ∈ RH with xα 6= xβ . So the properties in (b) are satisfied.

Theorem 4.18. Let X be a set with |X | > 2.

(a) If |X | ≤ ω, then S(X) ∈ T.(b) If |X | > c, then S(X) 6∈ T.

Proof. (a) Assume that 2 < |X | < ω. Since Z(S(X)) is trivial, S(X) ∈ T byExample 4.7(a).

Assume that |X | = ω. We can suppose X = Z. Let H = 〈σ, τ〉, whereτ = (−1, 1) and σ is the shift, that is σ(n) = n + 1 for every n ∈ Z. ThenOH(0) = Z and so RH = 0. Moreover S0 = 〈τ〉 and hence NH(S0) = S0. ByProposition 4.17 S(X) ∈ T.

(b) Let H be a finitely generated subgroup of S(X). Since OH(x) is count-able for every x ∈ RH , |RH | = |X | > c. Since H is countable, it has at most c

subgroups and so there exists a subset S of RH such that |S| > c and Sx = Sy

for every x, y ∈ S. By Proposition 4.17 S(X) 6∈ T.

Question 4.19. Let X be a set.

(a) Is S(X) ∈ T if |X | = ω1?(b) Is S(X) ∈ T if |X | = c?(c) Is S(X) ∈ T if ω < |X | ≤ c?

Remark 4.20. Question 4.19 can be formulated in equivalent terms thanksto Proposition 4.17. Indeed, if X is a set of cardinality κ with ω < κ ≤ c, thenS(X) ∈ T if and only if there exists a finitely generated group H admitting afamily S = Sα : α < κ of subgroups of H such that Sα = NH(Sα) for everyα < κ.

So Question 4.19 becomes: does there exist a finitely generated group Hwith a “large” (i.e., of cardinality κ with ω < κ ≤ c) family of self-normalizingsubgroups?

5. Problem A

We start considering a stability property of the the class A of Arnautovgroups.

Theorem 5.1. The class A is closed under taking quotients.


Proof. Let G ∈ A and let N be a normal subgroup of G. Let σ ≤ τ be grouptopologies on G/N such that idG/N : (G/N, τ) → (G/N, σ) is semitopological.Then idG : (G, τi) → (G, σi) is semitopological by Lemma 2.6. Since G ∈ A,τi = σi and hence τ = σ.

In Section 5.2 we will comment the stability of A under taking subgroupsand products.

Example 5.2. (a) Obviously every indiscrete group G is A-complete.(b) Let G be a group. Let Gab = G/G′ be the abelianization of G and

endow Gab with the discrete topology and with the indiscrete topology:

(G, ζG′ ) −−−−→ (Gab, δGab)

idG

yyidGab

(G, ιG) −−−−→ (Gab, ιGab)

If G 6= G′ then idGabis a semitopological non-open isomorphism, be-

cause Gab is abelian, and idG is a semitopological non-open isomor-phism too, in view of Remark 3.15. So (G, ζG′) is not A-complete.

(c) An abelian topological group G is A-complete if and only if G is indis-crete. In particular the only abelian Arnautov group is G = eG (as(G, δG) must be indiscrete).

The next proposition generalizes the example in (b).

Proposition 5.3. A topological group G with indiscrete derived group G′ isA-complete precisely when G is indiscrete.

Proof. The conclusion follows from Remark 3.15.

Example 5.4. Let G be a group and τ a group topology on G.

(a) If (G, τ) is SIN, then it is A-complete if and only if for every grouptopology σ < τ on G there exist U ∈ V(G,τ)(eG) and g ∈ G such that[g, Vg] 6⊆ U for every Vg ∈ V(G,σ)(eG) (this follows from Proposition2.8).

(b) If (G, τ) is Hausdorff and τ ≤ ζG′ (as already noted after Theorem2.7, this condition yields τ SIN), then G is abelian and consequentlyτ > ιG implies that (G, τ) is not A-complete (supposing that G is nota singleton).

Proposition 5.5. Let G be a group and N a normal subgroup of G. Let τbe a group topology on G/N and τi the initial topology of τ on G. Then τ isA-complete if and only if τi is A-complete.

Proof. Let idG/N : (G/N, τ) → (G/N, σ) be semitopological, where σ ≤ τ isanother group topology on G. By Lemma 2.6 also idG : (G, τi) → (G, σi) issemitopological and the hypothesis implies that τi = σi. Consequently τ = σ.


Suppose that τ is A-complete. Let σ < τi be another group topology on Gand consider the quotient topology σq of σ on G/N . So we have the followingsituation:

(G, τi)idG−−−−→ (G, σ)

π

yyπ

(G/N, τ)idG/N−−−−→ (G/N, σq).

Since σ < τi, it follows that Nσ ≥ Nτi = N . Consequently σ is the initialtopology of σq and so σq < τ , otherwise σ = τi. By hypothesis idG/N :(G/N, τ) → (G/N, σq) is not semitopological. To conclude that also idG :(G, τi) → (G, σ) is not semitopological apply Theorem 2.3.

Corollary 5.6. Let G be a group and τ a group topology on G. Considerthe quotient G/Nτ and the quotient topology τq of τ on G/Nτ . Then τ isA-complete if and only if τq is A-complete.

Proof. Since τ is the initial topology of τq, it suffices to apply Proposition5.5.

Now we give a necessary condition for a group to be Arnautov.

Proposition 5.7. For a group G the following conditions are equivalent:

(a) idG : (G, τ) → (G, ιG) is semitopological for no group topology τ > ιGon G;

(b) G is perfect.

Proof. (a)⇒(b) Since idG : (G, ζG′) → (G, ιG) is a semitopological isomorphismby Theorem 2.7(b), our hypothesis (a) implies ζG′ = ιG and hence G = G′.

(b)⇒(a) Suppose G = G′; then ζG′ = ιG. If idG : (G, τ) → (G, ιG) isa semitopological isomorphism, then τ ≤ ζG′ = ιG by Theorem 2.7(b), soτ = ιG. This means that idG is open.

Therefore, if a group G is Arnautov, then for every non-indiscrete grouptopology τ on G idG : (G, τ) → (G, ιG) is not semitopological. In particularProposition 5.7 implies that every Arnautov group is perfect.

Corollary 5.8. Let G be a simple non-abelian group and τ a group topologyon G. If τ > ιG, then idG : (G, τ) → (G, ιG) is not semitopological.

A consequence of these results is that every minimal simple non-abeliangroup (G, τ) is A-complete. Indeed, if σ ≤ τ is another group topology on Gand idG : (G, τ) → (G, σ) is semitopological, then by Lemma 3.2 either σ isHausdorff or σ = ιG. Since G is simple and non-abelian, G is perfect. ThenProposition 5.7 implies that σ is not indiscrete and so σ has to be Hausdorff.The minimality of τ yields that σ = τ .

This consequence is improved by the next result.

Proposition 5.9. If (G, τ) is a minimal group and Z(G) = eG, then (G, τ)is A-complete.


Proof. Let σ ≤ τ be a group topology on G and suppose that idG : (G, τ) →(G, σ) is semitopological. By Proposition 2.11 σ is Hausdorff and so σ = τ bythe minimality of τ .

Example 5.10. Every simple finite non-abelian group G is an Arnautov group.Indeed, the only group topologies on G are the trivial ones and idG : (G, δG) →(G, ιG) is not semitopological by Corollary 5.8.

The following remark could be used as a test to verify if a group is Arnautov.

Remark 5.11. If G ∈ A, then for every group topology τ on G and for everynormal subgroup N of G,

• idG : (G, supτ, ζN) → (G, τ) is not semitopological if supτ, ζN > τ ;• idG : (G, supτ, ζN) → (G, ζN ) is not semitopological if supτ, ζN >

ζN .

5.1. When the discrete topology is A-complete.

Remark 5.12. [14, Corollary 5.3] We can formulate Theorem 2.7(a) in termsof the Taımanov topology:

Let G be a group and σ a group topology on G. Then idG :(G, δG) → (G, σ) is semitopological if and only if σ ≥ TG, thatis, Nσ ≤ NTG = Z(G).

Consequently the Taımanov topology is the coarsest topology σ on a groupG such that idG : (G, δG) → (G, σ) is semitopological. So, since in this sectionwe consider the case when the discrete topology is A-complete, we have toimpose that the Taımanov topology is discrete, that is, the group is Taımanov.This also motivates Definition 1.8.

The next theorem solves a particular case of Problem A, that is, it charac-terizes the groups for which the discrete topology is A-complete.

Theorem 5.13. Let G be a group. Then δG is A-complete if and only if G ∈ T.

Proof. Suppose that δG > TG. Then idG : (G, δG) → (G, TG) is semitopologicalby Remark 5.12. This proves that δG is not A-complete. Suppose that δG = TG.Let τ < δG be a group topology on G. Then idG : (G, δG) → (G, τ) is notsemitopological by Remark 5.12. This proves that δG is A-complete.

By Proposition 4.5(a) the equivalent conditions of this theorem imply thatthe group has trivial center. The next example shows that they can be strictlystronger than having trivial center. Moreover this is an example of a Taımanovgroup which has an infinite non-abelian subgroup that is not Taımanov.

Example 5.14. Consider S(N) and let G := Sω(N) be the subgroup of S(N)of the permutations with finite support, that is Sω =

⋃∞n=1 Sn. Then Z(G) =

eG. If F is a finite subset of G, then there exists n ∈ N+ such that F ⊆ Sn

and c(Sn) = S(N \ 1, . . . , n) is infinite. Therefore TG < δG and so G 6∈ T.


Anyway in the finite case the three conditions are equivalent, as stated byExample 4.7(a).

The next theorem characterizes the almost trivial topologies that are A-complete. It covers Theorem 5.13.

Theorem 5.15. Let G be a group and N ⊳ G. Then (G, ζN ) is A-complete ifand only if G/N ∈ T.

Proof. Suppose that ζN is A-complete. Since ζN is the initial topology of δG/N ,it follows that δG/N is A-complete by Proposition 5.5. By Theorem 5.13 thisis equivalent to G/N ∈ T.

Suppose now that G/N ∈ T. By Theorem 5.13 this is equivalent to say thatδG/N is A-complete and so ζN is A-complete by Corollary 5.6.

The next theorem offers a relevant necessary condition for a group to beArnautov:

Theorem 5.16. If G ∈ A, then G ∈ Tt.

Proof. The conclusion follows from Theorems 5.1 and 5.13.

So the next question naturally arises.

Question 5.17. Does G ∈ Tt imply G ∈ A?

We shall give a positive answer to this question in a particular case in Propo-sition 5.25.

The next examples show that a group can admit two A-complete topologiesthat are one strictly finer than the other.

Example 5.18. Let G := S(Z) and S := Sω(Z) > A := Aω(Z), which are theonly proper normal subgroups of G.

(a) The point-wise convergence topology T on G is A-complete: T is min-imal and Z(G) is trivial, so Proposition 5.9 applies.

(b) The discrete topology δG is A-complete by Theorems 4.18 and 5.13.(c) We show that Z(G/A) = S/A and |S/A| = 2. The group S/A has only

one non-trivial element, that is, S/A = 〈π(τ)〉, where π : G → G/Ais the canonical projection and τ = (12) ∈ G. Indeed, if σ ∈ S andσ 6∈ A, then τσ ∈ A and so π(σ) ∈ 〈π(τ)〉. Moreover τ 6∈ A. SinceS/A is a non-trivial normal subgroup of G/A and it has size 2, it iscentral; since S/A is the unique non-trivial normal subgroup of G/A,S/A = Z(G/A).

(d) It follows from (c) that G 6∈ T by Proposition 4.5(a).(e) By (d) ζA is not A-complete in view of Theorem 5.15, hence G 6∈ A.(f) Moreover it is possible to prove that G/S ∈ T. Consequently G/S ∈ Tt,

being simple.

This is an example of a group G which is not Arnautov but with δG A-complete. Moreover, since the subgroup of G generated by the shift σ is abelian


and so not A-complete, while δG is A-complete, this example shows also thata subgroup of an A-complete group need not be A-complete.

Example 5.19. Consider the group G := SO3(R). As shown by Example 4.14,G ∈ T. Consequently δG is A-complete by Theorem 5.13. Moreover the usualcompact topology τ of G is A-complete, because τ compact implies minimal,Z(G) is trivial and so Proposition 5.9 applies.

A first step to find an answer to Question 5.17 is to consider the following.

Question 5.20. (a) Does S(Z)/Sω(Z) ∈ A?(b) Does SO3(R) ∈ A?

5.2. Totally Markov groups. Our aim is to provide examples of groups inA.

The next results shows that for totally Markov groups the topologies are allalmost trivial and so to verify if a continuous isomorphism of a totally Markovgroup is semitopological is simple, thanks to Corollary 3.13.

Proposition 5.21. A group G ∈ Mt if and only if every group topology on Gis almost trivial.

Proof. Suppose that G ∈ Mt and let τ be a group topology on G. Then thequotient topology of τ on G/Nτ is Hausdorff and hence discrete, being G ∈ Mt.So Nτ is open in (G, τ) and therefore τ is almost trivial.

Suppose that the group G 6∈ Mt. Then there exists a normal subgroup N ofG such that there exists a Hausdorff non-discrete group topology σ on G/N .Let π : G → G/N be the canonical projection and τ = π−1(σ). ThereforeNτ = N (because N =

⋂V : V ∈ V(G/N,σ)(eG/N ) in G/N). Since σ is

non-discrete N is not open and so τ is not almost trivial.

Proposition 3.10, together with Proposition 5.21, immediately implies thatMt is closed under extensions:

Definition 5.22. For a class of abstract groups P one says that P is closedunder extensions, if a group G belongs to P whenever N ∈ P and G/N ∈ Pfor some normal subgroup N of G.

Moreover we have the same result for M:

Theorem 5.23. The classes M and Mt are closed under extensions. In par-ticular, M and Mt are closed under finite direct products.

Proof. That Mt is closed under extensions is a direct consequence of Proposi-tions 3.10 and 5.21.

Suppose that the group G has a normal subgroup N such that N ∈ M andG/N ∈ M. We show that G ∈ M. To this end let τ be a Hausdorff grouptopology on G. Then τ N= δN . Consequently:

(i) N is closed in (G, τ), and(ii) π : (G, τ) → (G/N, τq) is a local homeomorphism.


By (i) (G/N, τq) is Hausdorff and so discrete. In view of (ii) τ = δG.

In view of Theorem 5.13, a necessary condition for A-completeness of δG fora group G is Z(G) = eG. For Markov groups also the converse implicationholds:

Corollary 5.24. Let G ∈ M. Then G ∈ T if and only if Z(G) = eG.

Proof. If G ∈ T, apply Theorem 5.13.Suppose Z(G) = eG. Then TG is Hausdorff by Lemma 4.1(b) and so

TG = δG.

In the following proposition we characterize totally Markov groups whichare A-complete or Arnautov. In particular it shows that for a totally Markovgroup it is equivalent to be Arnautov and to be totally Taımanov, which isprecisely the answer to Question 5.17 in the particular case of totally Markovgroups.

Proposition 5.25. Let G ∈ Mt.

(a) If τ is a group topology on G, the following conditions are equivalent:(i) (G, τ) is A-complete;(ii) G/Nτ ∈ T;(iii) for every N ⊳ G, if [G, N ] ≤ Nτ ≤ N , then N = Nτ .

(b) The following conditions are equivalent:(i) G ∈ A;(ii) G ∈ Tt;(iii) Z(G/N) = eG/N for every N ⊳ G;(iv) [G, N ] = N for every N ⊳ G.

Proof. (a) The equivalence (i)⇔(ii) follows from Lemma 5.21 and Theorem5.15. The equivalence (i)⇔(iii) follows from Lemma 5.21 and Corollary 3.13.

(b) The equivalence (i)⇔(ii) follows from (a) and the equivalence (ii)⇔(iii)follows from Corollary 5.24.

(iii)⇒(iv) Let N be a normal subgroup of G. Then [G, N ] is a normalsubgroup of G and Z(G/[G, N ]) is trivial by hypothesis. Since N/[G, N ] ≤Z(G/[G, N ]) also N/[G, N ] is trivial, that is N = [G, N ].

(iv)⇒(i) By Lemma 5.21 every group topology on G is almost trivial. So letL be a normal subgroup of G. For every normal subgroup N of G such that[G, N ] ≤ L ≤ N , N = L because [G, N ] = N by hypothesis. This proves thatζL is A-complete by (a). Consequently G ∈ A.

This proposition covers Example 5.10.

Corollary 5.26. (a) A finite group G ∈ A if and only if G ∈ Tt.(b) For every G ∈ M simple, G ∈ A.

In Example 5.18 we have seen that S(Z) 6∈ A, but S(Z)/Sω(Z) ∈ Tt. Inrelation to Question 5.20 we consider the following, which has also its owninterest. In Example 4.14 we have seen that SO3(R) ∈ Tt, but clearly SO3(R) 6∈M.


Question 5.27. Does S(Z)/Sω(Z) ∈ M?

A positive answer to this question would imply that S(Z)/Sω(Z) ∈ A,that is a positive answer to Question 5.20, in view of Corollary 5.26(b), sinceS(Z)/Sω(Z) is simple. From another point of view, in order to answer Ques-tion 5.27, it is possible to consider first Question 5.20 which involves a weakercondition.

Example 5.28. Let V = (Fpm)n, where m, n ∈ N+, p ∈ P and (n, pm−1) = 1.Define G to be the semidirect product of SL(V ) and V . Then [G, V ] = V .Moreover every normal subgroup of G contains V and so, since SL(V ) is simple,V is the unique non trivial normal subgroup of G. Then G ∈ A by Corollary5.26(a).

Example 5.29. (a) Corollary 5.26(b) provides an example of an infiniteArnautov group. Indeed Shelah [24] constructed a simple Markov(hence totally Markov) group M under CH.

(b) The group M contains a subgroup isomorphic to Z, which is abelianand so not in A.

(c) In general a totally Markov group need not be an Arnautov group, thatis, Mt 6⊆ A; for example G := M × Z(2) ∈ Mt but G 6∈ A.

Item (b) of this example shows that A is not stable under taking subgroups.

Question 5.30. Is A stable under taking (finite) direct products? And undertaking (finite) powers?

In the next example we give examples of Arnautov groups which are notsimple. Moreover we see a particular case (that of Markov simple groups) inwhich finite powers of Arnautov groups are Arnautov.

Example 5.31. Let M ∈ M be simple; by Corollary 5.26(b) M ∈ A. We showthat Mn ∈ Mt and also Mn ∈ A, for every n ∈ N+.

Since M ∈ M is simple, M ∈ Mt. By Theorem 5.23 Mn ∈ Mt for everyn ∈ N+. So Mn ∈ A by Proposition 5.25(b): for every normal subgroup N ofMn, N = Mk for some k ≤ n up to topological isomorphisms, and consequently[Mn, N ] = [Mn, Mk] = Mk = N .

The next are corollaries of Propositions 3.10 and 5.21.

Corollary 5.32. Let G be a group and N1 ≤ N2 be normal subgroups of Gwith N2/N1 ∈ Mt. Then every group topology τ on G with ζN2

≤ τ ≤ ζN1is

almost trivial. In particular,

(a) if N2 ∈ Mt, then every group topology τ on G with τ ≥ ζN2is almost

trivial; and(b) if G/N1 ∈ Mt, then every group topology τ on G with τ ≤ ζN1

is almosttrivial.

Proof. (a) Since N2 ∈ Mt, by Proposition 5.21 τ N2is almost trivial. Moreover

τq ≥ (ζN2)q = δG/N2

on G/N2, and so τq = δG/N2and in particular it is almost

trivial. By Proposition 3.10 τ is almost trivial.


Obviously, N1 ≤ Nτ ≤ N2. Therefore, the quotient topology τq of (G, τ)with respect to N1 satisfies δG/N1

≥ τq ≥ ζN2/N1. To the normal subgroup

N2/N1 ∈ Mt of the group G/N1 and τq ≥ ζN2/N1we apply (a) to claim that

τq is almost trivial. Since τq was obtained from τ via a quotient with respectto the τ -indiscrete normal subgroup N1, by Lemma 3.8 τ is almost trivial.

(b) Follows from the previous part.

Corollary 5.33. Let G be a group and N1 ≤ N2 be normal subgroups of Gwith [N2 : N1] finite. Then G admits only finitely many group topologies τ withζN2

≤ τ ≤ ζN1and they are all almost trivial.

Proof. Apply Corollary 5.32 to conclude that every group topology τ on Gsuch that ζN2

≤ τ ≤ ζN1is almost trivial. Moreover these τ are finitely many

because [N2 : N1] is finite.

Remark 5.34. A group G is hereditarily non-topologizable in case every sub-group of G is totally Markov [18]. Thus

hereditarily non-topologizable ⇒ totally Markov ⇒ Markov.

Consequently every group topology on a hereditarily non-topologizable groupis almost trivial.

If a hereditarily non-topologizable group G is Arnautov, then every quotientof G is Arnautov.

While infinite Arnautov groups exist (see Example 5.29(a)), it is not knownif there exists any infinite non-topologizable group. The existence of such agroup would solve an open problem from [10].

6. Problem B

We start by underlying an important aspect of AK-completeness comparedto K-minimality, where K is a class of topological groups. Indeed, let us re-call first that AG-completeness coincides with A-completeness and implies AK-completeness (see Remark 1.10). The K-minimal groups are precisely the in-discrete groups, whenever K contains all indiscrete groups. This fails to be truefor AK-completeness. In fact, the group G = S(Z), equipped with either thediscrete or the pointwise convergence topology, is A-complete (so AK-complete,for every K ⊆ G) as shown by Example 5.18(a,b). More generally for every non-trivial G ∈ T, the (obviously) non-indiscrete group (G, δG) is A-complete (soAK-complete, for every K ⊆ G) by Theorem 5.13.

As we have seen in Section 5 A-complete (i.e., AG-complete) groups are noteasy to come by. In order to have a richer choice of groups, we consider AK-complete groups for appropriate subclasses K of G. In case the subclass K iscompletely determined by an algebraic property (i.e., for every group topology τon G, (G, τ) ∈ K if and only if (G, δG) ∈ K), then obviously a topological group(G, τ) ∈ K is AK-complete if and only if it is A-complete. A typical exampleto this effect is the class of all topological abelian groups, or more generallythe class of all topological groups such that the underlying group belongs to a


fixed variety V (in the sense of [20]) of abstract groups. We formulate an openquestion for a specific V in Question 6.13.

In the sequel we consider subclasses K ⊆ G of a different form, most oftenK ⊆ H.

Since H-minimality coincides with minimality, AH-completeness is a gener-alization of minimality. It is a strict generalization in view of (a) of the nextexample.

Example 6.1.

(a) The group (S(Z), δS(Z)) is A-complete, as shown by Example 5.18(b),and consequently AH-complete, but it is not minimal: δS(Z) and thepoint-wise convergence topology T are both Hausdorff.

(b) Let G ∈ T be non-torsion. Then (G, δG) is A-complete by Theorem5.13, and in particular it is AH-complete. On the other hand, by ourhypothesis there exists x ∈ G of infinite order, that is 〈x〉 is abelianand so not AH-complete. This shows that in general a subgroup ofan AH-complete group need not be AH-complete. (This is noted afterExample 5.18 for the particular case of (S(Z), δS(Z)).)

Anyway AH-completeness coincides with minimality in the abelian case:

Proposition 6.2. If G is an abelian group and (G, τ) ∈ H, then (G, τ) isAH-complete if and only if it is minimal.

This proposition gives a partial answer to Problem B for the subclass of Hof abelian topological groups. The problem remains open for the larger classH:

Question 6.3. When is a topological group (G, τ) ∈ H AH-complete? And inwhich cases is (G, δG) AH-complete?

The next example, that extends Example 5.18(a), motivates Lemma 6.5.

Example 6.4. For an infinite topologically simple (i.e., there exists no non-trivial closed normal subgroup) Hausdorff non-abelian group (G, τ), minimalimplies A-complete. In fact Z(G) = eG and Proposition 5.9 applies.

The next lemma and corollary provide partial answers to Question 6.3.Lemma 6.5 in particular covers the previous example, since it implies thatevery minimal group with trivial center is A-complete (in view of the fact thatminimal implies AH-minimal).

Lemma 6.5. Let G be a group with Z(G) = eG and let τ be a Hausdorffgroup topology on G. Then (G, τ) is AH-complete if and only if (G, τ) is A-complete.

Proof. If (G, τ) is A-complete, then it is AH-complete.Suppose that (G, τ) is AH-complete. Let σ ≤ τ be a group topology on G

such that idG : (G, τ) → (G, σ) is semitopological. By Proposition 2.11 σ isHausdorff. Then σ = τ . This proves that (G, τ) is A-complete.


This lemma implies Proposition 5.9, since minimal groups are AH-complete.

Corollary 6.6. Let G be a group. Then Z(G) = eG and δG is AH-completeif and only if G ∈ T.

Proof. If Z(G) = eG and δG is AH-complete, then δG is A-complete byLemma 6.5 and so G ∈ T by Theorem 5.13.

Assume that G ∈ T. By Theorem 5.13 δG is A-complete and so AH-complete. Moreover Z(G) = eG by Proposition 4.5(a).

Lemma 6.5 suggests the following question: is Z(G) = eG a necessarycondition for the validity of the implication (G, τ) AH-complete ⇒ (G, τ) A-complete? According to Corollary 6.6 the answer is “yes” in case τ is thediscrete topology.

Proposition 6.7. Let (G, τ) be a SIN Hausdorff group. If (G, τ) is A-complete,then Z(G) = eG.

Proof. Suppose that Z(G) 6= eG. We want to see that (G, τ) fails to beA-complete. Consider the topology T := τ ∧ ζZ(G), which has as a local baseat eG the family BT = U · Z(G) : U ∈ V(G,τ)(eG). Since τ is Hausdorff andT is not Hausdorff (because Z(G) 6= eG), τ > T . So it remains to prove thatidG : (G, τ) → (G, T ) is semitopological. Since (G, τ) is SIN, it suffices to provethat for every U ∈ V(G,τ)(eG) and for a fixed g ∈ G there exists Vg ∈ BT suchthat [g, Vg] ⊆ U and then apply Proposition 2.8. So let U ∈ V(G,τ)(eG) andg ∈ G. Since (G, τ) is SIN, there exists U ′ ∈ V(G,τ)(eG) such that U ′U ′ ⊆ U

and gU ′g−1 ⊆ U ′. Let Vg = U ′ ·Z(G) ∈ BT . Then [g, Vg] = [g, U ′] ⊆ U ′U ′ ⊆ U .Since we have proved that idG : (G, τ) → (G, T ) is semitopological and τ > T ,then (G, τ) fails to be A-complete.

Remark 6.8. As a consequence of Lemma 6.5 and Proposition 6.7 we havethe following equivalence between A-completeness and the purely algebraicproperty of having trivial center. Indeed, if (G, τ) ∈ H is AH-complete, thenZ(G) = eH implies (G, τ) A-complete by Lemma 6.5. Moreover, if (G, τ)is SIN, in view of Proposition 6.7 also the converse implication holds, that is,(G, τ) is A-complete if and only if Z(G) = eG.

Corollary 6.9. Let (G, τ) be a Hausdorff group with Z(G) 6= eG.

(a) If (G, τ) is SIN and AH-complete, then it is not A-complete.(b) If (G, τ) is SIN and minimal, then it is not A-complete.(c) If (G, τ) is compact, then it is not A-complete.

This corollary produces in particular examples of AH-complete groups whichare not A-complete (e.g., compact groups with non-trivial center), showing thatthe implication (G, τ) AH-complete ⇒ (G, τ) A-complete may fail to be true,also for non-discrete groups. In particular in Example 6.12 shows a group, withnon-trivial center, which does not admit any compact topology, but admitsminimal linear (so SIN) topologies, that are not A-complete by Corollary 6.9.


Proposition 6.10. Let G be a group such that G ∈ T and let F be a finitegroup. Then δG×F is AH-complete.

Proof. Let τ be a Hausdorff group topology on G×F and suppose that idG×F :(G × F, δG×F ) → (G × F, τ) is semitopological. By Remark 5.12 τ ≥ TG×F .But TG×F = TG × TF = δG × TF by Lemma 4.4. So τ ≥ δG × TF . Since τ isHausdorff, τ = δG×F , and this proves that δG×F is AH-complete.

Using this proposition we can give examples of AH-complete groups whichare not A-complete, as the following. Another example of an AH-completegroup which is not A-complete is in Example 6.12.

Example 6.11. Let G = S(Z) × Z(2). By Theorem 4.18(a) S(Z) ∈ T. Then(G, δG) is AH-complete by Proposition 6.10. Since Z(G) = idZ×Z(2) is nottrivial, G 6∈ T by Proposition 4.5(a). Consequently G is not A-complete byTheorem 5.13.

Example 6.12. Let p ∈ P and let G be the group HZ (see Example 4.3)equipped with the product topology T = P (τp, τp, τp) where τp is the p-adictopology of Z. A base of T is given by the family of the (normal) subgroups

formed by the matrices of the form

1 pnZ pnZ

0 1 pnZ

0 0 1

. Clearly G is SIN. Then

(G, T ) is minimal [5, 7], so AH-complete. Moreover (G, T ) is A-complete byCorollary 6.9.

Considering SIN groups in Example 5.4, Proposition 6.7 and Corollary 6.9we have weakened the commutativity from a topological point of view. A dif-ferent way to weaken commutativity, but algebraically, is to consider nilpotenttopological groups:

Question 6.13. If (G, τ) is a nilpotent topological group, when is (G, τ) A-complete?

The following example is dedicated to a very particular case of this question.

Example 6.14. Consider the class

K0R

:= (HR, P (τ, τ, τ)) : τ is a ring topology on R,

where P (τ, τ, τ) denotes the product topology on G. Then every G ∈ K0R

isAK0

R

-complete.

Indeed, let τ ≥ σ be ring topologies on R such that

(HR, P (τ, τ, τ)), (HR, P (σ, σ, σ)) ∈ K0R.

Suppose that idR : (HR, P (τ, τ, τ)) → (HR, P (σ, σ, σ)) is semitopological. By

Theorem 1.2, for every U ′ =

1 U U0 1 U0 0 1

∈ V(HR,P (τ,τ,τ))(eHR

) and h =


1 1 00 1 00 0 1

there exists Vh =

1 V V0 1 V0 0 1

∈ V(HR,P (σ,σ,σ))(eHR) such that

[h, Vh] ⊆ U ′. In particular this implies V ⊆ U and hence σ ≥ τ , that is σ = τ .

In a forthcoming paper [6] we extend this result to the more general case ofgeneralized Heisenberg groups on an arbitrary unitary ring A.

7. Problem C

Problem C is about compositions of semitopological isomorphisms. In orderto measure more precisely the level of being semitopological, we introduce thenext notion.

Definition 7.1. Let G be a group, σ ≤ τ group topologies on G and n ∈ N+.Then idG : (G, τ) → (G, σ) is n-step semitopological if there exist n − 1 grouptopologies σ ≤ λn−1 ≤ . . . ≤ λ1 ≤ τ on G such that idG : (G, τ) → (G, λ1), idG :(G, λ1) → (G, λ2), . . . , idG : (G, λn−1) → (G, σ) are semitopological.

Obviously idG : (G, τ) → (G, σ) is 1-step semitopological if and only if it issemitopological. Moreover a continuous isomorphism of topological groups iscomposition of semitopological isomorphisms if and only if it is n-step semi-topological for some n ∈ N+.

Let G be a non-trivial group. The lower central series of G is defined byγ1(G) = G and γn(G) = [G, γn−1(G)] for every n ∈ N, n ≥ 2. The uppercentral series of G is defined by Z0(G) = eG, Z1(G) = Z(G) and Zn(G)is such that Zn(G)/Zn−1(G) = Z(G/Zn−1(G)) for every n ∈ N, n ≥ 2. Agroup G is nilpotent if and only if γn(G) = eG for some n ∈ N+, if andonly if Zm(G) = G for some m ∈ N+. The minimum n ∈ N+ such thatγn+1(G) = eG, that is, the minimum n ∈ N+ such that Zn(G) = G, is theclass of nilpotency of G.

Our main theorem about n-step semitopological isomorphisms is the follow-ing. It is an answer to Problem C(a) in the particular case when the topologieson the domain and on the codomain are the discrete and the indiscrete onerespectively.

Theorem 7.2. Let G be a group and n ∈ N+. Then idG : (G, δG) → (G, ιG)is n-step semitopological if and only if G is nilpotent of class ≤ n.

Proof. If idG : (G, δG) → (G, ιG) is n-step semitopological, then there existn − 1 group topologies λn−1 ≤ . . . ≤ λ1 on G such that

idG : (G, δG) → (G, λ1), idG : (G, λ1) → (G, λ2), . . .

. . . , idG : (G, λn−2) → (G, λn−1), idG : (G, λn−1) → (G, ιG)

are semitopological. By Theorem 2.7(b) G′ ⊆ V for every V ∈ V(G,λn−1)(eG).Since idG : (G, λn−2) → (G, λn−1) is semitopological, Theorem 1.2 implies thatfor every U ∈ V(G,λn−2)(eG) and for every g ∈ G there exists Vg ∈ V(G,λn−1)(eG)such that [g, Vg] ⊆ U . Consequently [g, G′] ⊆ U for every U ∈ V(G,λn−2)(eG).


Hence γ3(G) = [G, G′] ⊆ U for every U ∈ V(G,λn−2)(eG). Proceeding byinduction we have that γn(G) ⊆ U for every U ∈ V(G,λ1)(eG). By Theorem2.7(a) cG(g) is λ1-open for every g ∈ G. Thus γn(G) ⊆ Z(G) and this impliesthat G is nilpotent of class ≤ n (γn+1(G) = eG).

Conversely, if G is nilpotent of class ≤ n, consider on G the group topologiesζZ(G), ζZ2(G), . . . , ζZn−1(G). Then idG : (G, δG) → (G, ζZ(G)) is semitopologicalby Theorem 2.7(a) and idG : (G, ζZn−1(G)) → (G, ιG) is semitopological becauseG′ ≤ Zn−1(G) since G/Zn−1(G) is abelian and applying Theorem 2.7(b). Forevery i = 1, . . . , n − 1, by Corollary 3.13 idG : (G, ζZi(G)) → (G, ζZi+1(G)) issemitopological if and only if [G, Zi+1(G)] ≤ Zi(G) and this holds true sinceZi+1(G)/Zi(G) = Z(G/Zi(G)).

As a particular case of n = 2 in this theorem, we find [2, Example 12], whichwitnesses that the composition of semitopological isomorphisms is not semi-topological in general. Indeed idG : (G, δG) → (G, ιG) is not semitopological,whenever G is not abelian.

For n ∈ N+, let

n-S := fn . . . f1 : fi ∈ S.

Observe that

S = 1-S ⊂ 2-S ⊂ . . . ⊂ n-S ⊂ (n + 1)-S ⊂ . . . ,

where all inclusions are proper by the previous theorem.Define also ∞-S :=

⋃∞n=1 n-S and observe that it is closed under compo-

sitions. Moreover ∞-S is closed also under taking subgroups, quotients andfinite products, in the following sense:

Lemma 7.3. Let n ∈ N+, let G be a group and τ ≥ σ group topologies on Gsuch that idG : (G, τ) → (G, σ) is n-step semitopological.

(a) If A is a subgroup of G, then idG A= idA : A → A is n-step semitopo-logical.

(b) If A is a normal subgroup of G, then idG/A : (G/A, τq) → (G/A, σq) isn-step semitopological.

Proof. (a) By hypothesis there exist n − 1 group topologies σ ≤ λn−1 ≤ . . . ≤λ1 ≤ τ on G such that

idG : (G, τ) → (G, λ1), idG : (G, λ1) → (G, λ2), . . . , idG : (G, λn−1) → (G, σ)

are semitopological. Theorem 2.3(a) implies that

idA : (A, τ A) → (A, λ1 A), idA : (A, λ1 A) → (A, λ2 A), . . .

. . . , idA : (A, λn−1 A) → (A, σ A)

are semitopological and so idA : (A, τ A) → (A, σ A) is n-step semitopologi-cal.

(b) Follows from Theorem 2.3(b).


The following lemma shows that for each n ∈ N+ the class n-S is closedunder taking products. In particular it implies that ∞-S is closed under takingfinite products.

Lemma 7.4. Let n ∈ N+, let Gi : i ∈ I be a family of groups and τi :i ∈ I, σi : i ∈ I two families of group topologies such that σi ≤ τi aregroup topologies on Gi and idGi : (Gi, τi) → (Gi, σi) is n-step semitopologicalfor every i ∈ I. Then

∏i∈I idGi :

∏i∈I(Gi, τi) →

∏i∈I(Gi, σi) is n-step

semitopological.

Proof. It follows from Theorem 2.4.

The following example shows that ∞-S is not closed under taking infinitedirect products and answers negatively (b) of Problem C. In fact we constructa continuous isomorphism which is not composition of semitopological isomor-phisms.

Example 7.5. For every n ∈ N+ let Gn be a nilpotent group of class n.Then

∏∞n=1 idGn :

∏∞n=1(Gn, δGn) →

∏∞n=1(Gn, ιGn) is n-step semitopological

for no n ∈ N+. Indeed idGn+1: (Gn+1, δGn+1

) → (Gn+1, ιGn+1) is not n-step

semitopological whenever n ∈ N+, in view of Theorem 7.2, because Gn+1 isnot nilpotent of class ≤ n.

The next example is another particular case in which we answer ProblemC(a).

Example 7.6. Let n ∈ N+, let G be a totally Markov group and τ, σ grouptopologies on G. Every group topology on G is almost trivial by Proposition5.21. Then idG : (G, τ) → (G, σ) is n-step semitopological if and only if[G, [G, [...[G︸︷︷︸

n

, Nσ]]]] ≤ Nτ .

In fact, suppose that idG : (G, τ) → (G, σ) is n-step semitopological. Thenthere exist group topologies σ ≤ λn−1,≤ . . . ,≤ λ1 ≤ τ on G such that

idG : (G, τ) → (G, λ1), idG : (G, λ1) → (G, λ2), . . .

. . . , idG : (G, λn−1) → (G, σ)

are semitopological. By Corollary 3.13

[G, Nσ] ⊆ Nλ1, [G, Nλ1

] ⊆ Nλ2, . . . , [G, Nλn−1

] ⊆ Nτ

and hence [G, [G, [...[G︸︷︷︸n

, Nσ]]]] ≤ Nτ .

Assume that [G, [G, [...[G︸︷︷︸n

, Nσ]]]] ≤ Nτ . Let

Nλ1= [G, Nσ], Nλ2

= [G, Nλ1], . . . , Nλn−1

= [G, Nλn−2].

By Corollary 3.13 and our assumption idG : (G, τ) → (G, λ1), idG : (G, λ1) →(G, λ2), . . . , idG : (G, λn−1) → (G, σ) are semitopological.


References

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in topological groups, Topology Appl., to appear.[8] D. Dikranjan, I. Prodanov and L. Stoyanov, Topological Groups: Characters, Duali-

ties and Minimal Group Topologies, Pure and Applied Mathematics, Vol. 130, MarcelDekker Inc., New York-Basel, 1989.

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68 (1977), 411–423.[16] T. Husain, Introduction to topological groups, Saunders, Philadelphia, 1966.[17] H. Kowalski, Beitrage sur topologischen albegra, Math. Naschr. 11 (1954), 143–185.[18] G. Lukacs, Hereditarily non-topologizable groups, arXiv:math/0603513v1 [math.GR].[19] M. Megrelishvili, Generalized Heisenberg groups and Shtern’s question, Georgian Math.

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x+192 pp.[21] A. Yu. Ol′shanskii, A remark on a countable non-topologized group, Vestnik Moskov

Univ. Ser. I Mat. Mekh. (1980), 103 (in Russian).[22] V. Ptak, Completeness and the open mapping theorem, Bull. Soc. Math. France 86

(1958), 41–74.[23] D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, Berlin, 1982.[24] S. Shelah, On a problem of Kurosh, Jonsson groups and applications, Word problems,

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Received July 2008

Accepted October 2008

Dikran Dikranjan ([email protected])Dipartimento di Matematica e Informatica, Universita di Udine, via delle Scienze,206 - 33100 Udine, Italy

Anna Giordano Bruno ([email protected])Dipartimento di Matematica e Informatica, Universita di Udine, via delle Scienze,206 - 33100 Udine, Italy



Volume 10, No. 1, 2009

pp. 121-130

New coincidence and common fixed pointtheorems

S. L. Singh, Apichai Hematulin and Rajendra Pant

Abstract. In this paper, we obtain some extensions and a gen-

eralization of a remarkable fixed point theorem of Proinov. Indeed,

we obtain some coincidence and fixed point theorems for asymptoti-

cally regular non-self and self-maps without requiring continuity and

relaxing the completeness of the space. Some useful examples and dis-

cussions are also given.

2000 AMS Classification: 54H25; 47H10.

Keywords: Coincidence point; fixed point; Banach contraction; quasi-contraction; asymptotic regularity.

1. Introduction

The well-known Banach fixed point theorem has been generalized and ex-tended by many authors in various ways. Recently, Proinov [15] has obtainedtwo types of generalizations of Banach’s fixed point theorem. The first typeinvolves Meir- Keeler type conditions (see, for instance, Cho et al. [3], Jachym-ski [6], Lim [10], Matkowski [11], Park and Rhoades [14]) and the second typeinvolves contractive gauge functions (see, for instance, Boyd and Wong [1] andKim et al. [9]). Proinov [15] obtained equivalence between these two types ofcontractive conditions and also obtained a new fixed point theorem. Inspiredby Jungck [7], Naimpally et al. [13], Proinov [15] and Romaguera [19], weobtain coincidence theorems on a very general setting and derive various fixedpoint theorems. Some special cases are also discussed.

In all that follows Y is an arbitrary non-empty set, (X, d) a metric space andN := 1, 2, 3, ..., . For T, f : Y → X , let C(T, f) denote the set of coincidencepoints of T and f , that is C(T, f) := z ∈ Y : Tz = fz.

The following definition comes from Sastry et al. [20] and S. L. Singh et al.[21].

122 S. L. Singh, A. Hematulin and R. Pant

Definition 1.1. Let S, T and f be maps on Y with values in a metric space(X, d). The pair (S, T ) is asymptotically regular with respect to f at x0 ∈ Y ifthere exists a sequence xn in Y such that

fx2n+1 = Sx2n, fx2n+2 = Tx2n+1, n = 0, 1, 2, ..., and

limn→∞

d(fxn, fxn+1) = 0.

If Y = X and S = T then we get the definition of asymptotic regularityof T with respect to f due to Rhoades et al. [18]. Further if Y = X , S = T

and f is the identity map on X , then we get the usual definition of asymptoticregularity for a map T due to Browder and Peteryshyn [2].

Definition 1.2 ([16]). Let (X, d) be a metric space and T, f : X → X. Thenthe self-maps T and f are R-weakly commuting if there exists a positive realnumber R such that

d(Tfx, fTx) ≤ Rd(Tx, fx) for all x ∈ X.

Following Itoh and Takahashi [5] and Singh and Mishra [22], we have thefollowing definition for a pair of self-maps on a metric space X .

Definition 1.3. Let T, f : X → X. Then the pair (T, f) is (IT)-commutingat z ∈ X if Tfz = fT z. They are (IT)-commuting on X (also called weaklycompatible, by Jungck and Rhoades [8]) if Tfz = fT z for all z ∈ X such thatTz = fz.

Definition 1.4 ([15] Definition 2.1 (i)). Let φ denote the class of all functionsϕ : R+ → R+ satisfying: for any ε > 0 there exists δ > ε such that ε < t < δ

implies ϕ(t) ≤ ε.

2. Main Results

Proinov [15] obtained the following result generalizing some fixed point the-orems of Jachymski [6] and Matkowski [11].

Theorem 2.1 ([15, Th. 4.1]). Let T be a continuous and asymptotically regularself-map on a complete metric space (X, d) satisfying the following conditions:

(P1): d(Tx, T y) ≤ ϕ(D(x, y)), for all x, y ∈ X;(P2): d(Tx, T y) < D(x, y), for all distinct x, y ∈ X,

where D(x, y) = d(x, y) + γ[d(x, Tx) + d(y, T y)], γ ≥ 0 and ϕ ∈ φ.

Then T has a unique fixed point.Moreover if D(x, y) = d(x, y) + d(x, Tx) + d(y, T y) and ϕ is continuous and

satisfies ϕ(t) < t for all t > 0, then continuity of T can be dropped.

For a self-map T : X → X the quasi-contraction due to Ciric [4] is as follows(C) d(Tx, T y) ≤ qM(x, y),where M(x, y) = maxd(x, y), d(x, Tx), d(y, T y), d(x, T y), d(y, Tx), 0≤ q < 1.

We remark that following the listing of conditions due to Rhoades [17] thecondition (C) is the condition (24). According to Rhoades [17] the condition(25):

New coincidence and common fixed point theorems 123

d(Tx, T y) < M(x, y),is the most general condition among the contractive conditions.

The following example shows that (P1) is more general than condition (C).

Example 2.2. Let X = 1, 2, 3 with the usual metric d and T : X → X suchthat

T 1 = 1, T 2 = 3, T 3 = 1. Then T satisfies (C) with q > 1.

Clearly, the condition (P1) is satisfied with ϕ(t) = t

2for all t > 0 and ϕ(0) = 0

and γ ≥ 1.

Evidently T can not satisfy the conditions (24) and (25) listed by Rhoades[17].

First we extend the scope of Theorem 2.1 by introducing a dummy map f

in Theorem 2.1. This idea comes essentially from Jungck [7].

We remark that the requirement “ϕ(t) < t for all t > 0” in Theorem 2.1 isredundant as this is the consequence of Definition 1.4. We shall use this factin the proof of the following theorem.

Theorem 2.3. Let T and f be self-maps on a complete metric space (X, d)such that

(A1): T (X) ⊆ f(X);(A2): d(Tx, T y) ≤ ϕ(g(x, y)) for all x, y ∈ X,

where g(x, y) = d(fx, fy) + γ[d(fx, Tx) + d(fy, T y)], γ ≥ 0 and ϕ ∈ φ

is continuous;(A3): d(Tx, T y) < g(x, y) for all distinct x, y ∈ Y ;(A4): (T, f) is asymptotically regular at x0 ∈ X.

If T is continuous then T has a fixed point provided that T and f are R-weaklycommuting. Further if f is continuous and γ = 1 then T and f have a uniquecommon fixed point provided that T and f are R-weakly commuting.

Proof. Pick x0 ∈ X . Define a sequence yn by yn+1 = Txn = fxn+1, n =0, 1, 2, ... This can be done since the range of f contains the range of T . Let usfix ε > 0. Since ϕ ∈ φ, there exists δ > ε such that for any t ∈ (0,∞),

(2.1) ε < t < δ ⇒ ϕ(t) ≤ ε.

Without loss of generality we may assume that δ ≤ 2ε. Since the pair (T, f)is asymptotically regular, lim

n→∞

d(yn, yn+1) = 0. Hence, there exists an integer

N ≥ 1 such that

(2.2) d(yn, yn+1) <δ − ε

1 + 2γfor all n ≥ N.

By induction we shall show that

(2.3) d(yn, ym) <δ + 2γε

1 + 2γfor all m, n ∈ N with m ≥ n ≥ N .


Let n ≥ N be fixed. Obviously, (2.3) holds for m = n. Assuming (2.3) to holdfor an integer m ≥ n, we shall prove it for m + 1. By the triangle inequality,we get

d(yn, ym+1) ≤ d(yn, yn+1) + d(yn+1, ym+1)

or

(2.4) d(yn, ym+1) ≤ d(yn, yn+1) + d(Txn, Txm).

We claim that

(2.5) d(Txn, Txm) ≤ ε.

To prove (2.5), we consider two cases.

Case 1.: Let g(xn, xm) ≤ ε. By (A2) and (A3),

d(Txn, Txm) ≤ g(xn, xm) ≤ ε, and (2.5) holds.

Case 2.: Let g(xn, xm) > ε. By (A2),

(2.6) d(Txn, Txm) ≤ ϕ(g(xn, xm)).

By the definition of g(x, y),

g(xn, xm) = d(yn, ym) + γ[d(yn, yn+1) + d(ym, ym+1)].

From (2.2) and (2.3),

g(xn, xm) <δ + 2γε

1 + 2γ+ 2γ

δ − ε

1 + 2γ= δ.

Now by (2.1),

ε < g(xn, xm) < δ ⇒ ϕ(g(xn, xm)) ≤ ε.

So (2.6) implies (2.5). From (2.5), (2.4) and (2.2), it follows that

d(yn, ym+1) ≤δ − ε

1 + 2γ+ ε =

δ + 2γε

1 + 2γ. This proves(2.3).

Since δ ≤ 2ε, (2.3) implies that d(yn, ym) < 2ε for all integers m and n withm ≥ n ≥ N . So yn is a Cauchy sequence. Since the space X is complete thesequence yn has a limit. Call it z.

Suppose T is continuous. Then TTxn → Tz and Tfxn → Tz. Since T andf are R-weakly commuting,

d(Tfxn, fTxn) ≤ Rd(Txn, fxn).

Making n → ∞,

fTxn → Tz. If z 6= Tz, then by (A2),

d(Txn, TTxn) ≤ ϕ(g(xn, Txn)

= ϕ(d(fxn, fTxn) + γ[d(fxn, Txn) + d(fTxn, TTxn)]).

Making n → ∞,

d(z, T z) ≤ ϕ(d(z, T z) < d(z, T z), a contradiction. It follows that z = Tz.


If f continuous and γ = 1. Then ffxn → fz and fTxn → fz. Since T andf are R-weakly commuting,

d(Tfxn, fTxn) ≤ Rd(Txn, fxn).

Making n → ∞,

Tfxn → fz. If z 6= fz, then by (A2),

d(Txn, T fxn) ≤ ϕ(g(xn, fxn)

= ϕ(d(fxn, ffxn) + γ[d(fxn, Txn) + d(ffxn, T fxn)]).

Making n → ∞,

d(z, fz) ≤ ϕ(d(z, fz) < d(z, fz), a contradiction. It follows that z = fz.

Now if z 6= Tz, then by (A2),

d(Tz, T fxn) ≤ ϕ(g(z, fxn)

= ϕ(d(fz, ffxn) + [d(fz, T z) + d(ffxn, T fxn)]).

Making n → ∞,

d(Tz, fz) ≤ ϕ(d(Tz, fz) < d(Tz, fz), a contradiction.

It follows that Tz = fz = z, and z is a common fixed point of f and T .Uniqueness follows easily.

We remark that Theorem 2.1 is obtained from Theorem 2.3 as a corollary.Notice that conditions (P1) and (P2) come respectively from (A2) and (A3)when f is the identity map on X . Further, the continuity of only one mapis needed. The following example shows the superiority of Theorem 2.3 overTheorem 2.1.

Example 2.4. Let X = [0,∞) with usual metric d. Let T : X → X such that

Tx =

x if x is rational,0 if x is irrational.

Theorem 2.1 is not applicable to this map T as it is not continuous. However,if we take a (dummy) map f : X → X such that fx = 2x for all x ∈ X then T

and f satisfy all the hypotheses of Theorem 2.3. Notice that f is continuousand T 0 = f0 = 0.

Now we modify certain requirements of Theorem 2.3 a slightly to obtain anew result.

Theorem 2.5. Let T and f be maps on an arbitrary non-empty set Y withvalues in a metric space (X, d) such that

(B1): T (Y ) ⊆ f(Y );(B2): d(Tx, T y) ≤ ϕ(g(x, y)) for all x, y ∈ Y ,

where g(x, y) = d(fx, fy) + γ[d(fx, Tx) + d(fy, T y)], 0 ≤ γ ≤ 1, andϕ : R+ → R+ continuous;


(B3): (T, f) is asymptotically regular at x0 ∈ Y .If T (Y ) or f(Y ) is a complete subspace of X then

(i): C(T, f) is non-empty.

Further, if Y = X, then(ii): T and f have a unique common fixed point provided that T and f

are (IT)-commuting at a point u ∈ C(T, f).

Proof. Pick x0 ∈ Y . Define a sequence yn by yn+1 = Txn = fxn+1, n =0, 1, 2..., this can be done since the range of f contains the range of T . Sincethe pair (f, T ) is asymptotically regular, lim

n→∞

d(yn, yn+1) = 0.

First we shall show that yn is a Cauchy sequence. Suppose yn is notCauchy. Then there exists µ > 0 and increasing sequences mk and nk ofpositive integers such that for all n ≤ mk < nk,

d(ymk, ynk

) ≥ µ and d(ymk, ynk−1) < µ.

By the triangle inequality,

d(ymk, ynk

) ≤ d(ymk, ynk−1) + d(ynk−1, ynk

).

Making k → ∞,

d(ymk, ynk

) < µ.

Thus, d(ymk, ynk

) → µ as k → ∞. Now by (B2),

d(ymk+1, ynk+1) = d(Txmk, Txnk

)

≤ ϕ(g(xmk, xnk

))

= ϕ(d(fxmk, fxnk

) + γ[d(fxmk, Txmk

) + d(fxnk, Txnk

)]).

Making k → ∞,

µ ≤ ϕ(µ) < µ,

a contradiction. Therefore yn is Cauchy. Suppose f(Y ) is complete. Thenyn being contained in f(Y ) has a limit in f(Y ). Call it z. Let u ∈ f−1z.Then fu = z. Using (B2),

d(Tu, Txn) ≤ ϕ(d(fu, fxn) + γ[d(Tu, fu) + d(Txn, fxn)]).

Making n → ∞,

d(Tu, z) ≤ ϕ(γd(Tu, z)) < d(Tu, z),

a contradiction. Therefore Tu = z = fu. This proves (i). Now if Y = X andthe pair(T, f) is (IT)-commuting at u then Tfu = fTu and TTu = Tfu =fTu = ffu. In view of (B2), it follows that

d(Tu, TTu) < ϕ(g(u, Tu))

= ϕ(d(fu, fTu) + γ[d(Tu, fu) + d(TTu, fTu)]) < d(Tu, TTu),

a contradiction. Therefore TTu = Tu and fTu = TTu = Tu = z. This proves(ii).


In the case T (Y ) is a complete subspace of X , the condition (B1) impliesthat sequence yn converges in f(Y ), and the previous proof works. Theuniqueness of common fixed point follows easily.

The following result generalizes an important result of Proinov [15, Cor. 4.3]

Corollary 2.6. Let T and f be maps on an arbitrary non-empty set Y withvalues in metric space (X, d) such that

(C1): T (Y ) ⊆ f(Y );(C2): d(Tx, T y) ≤ ϕ(M(x, y)), for all x, y ∈ Y ,

where M(x, y) = maxd(fx, fy), d(fx, Tx), d(fy, T y), 12[d(fx, T y)+

d(fy, Tx)] and ϕ : R+ → R+ continuous.If T (Y ) or f(Y ) is a complete subspace of X then conditions (i) and

(ii) of above Theorem 2.5 hold.

Now we obtain a new common fixed point theorem for three non self-maps.

Theorem 2.7. Let S, T and f be maps on an arbitrary non-empty set Y withvalues in a metric space (X, d). Let (S, T ) be asymptotically regular with respectto f at x0 ∈ Y and the following conditions are satisfied:

(D1): S(Y ) ∪ T (Y ) ⊆ f(Y );(D2): d(Sx, T y) ≤ ϕ(h(x, y)), for all x, y ∈ X,

where h(x, y) = d(fx, fy) + γ[d(Sx, fx) + d(Ty, fy)], 0 ≤ γ ≤ 1, andϕ : R+ → R+ continuous.

If S(Y ) or T (Y ) or f(Y ) is a complete subspace of X then(I): C(S, f) is non-empty;(II): C(T, f) is non-empty.

Further, if Y=X then(III): S and f have a common fixed point provided that S and f are

(IT)-commuting at a point u ∈ C(S, f).(IV): T and f have a common fixed point provided that T and f are

(IT)-commuting at a point v ∈ C(T, f).(V): S, T and f have a unique common fixed point provided that (III)

and (IV) both are true.

Proof. Let x0 be an arbitrary point in Y . Since (S, T ) is asymptotically regularwith respect to f , then there exists a sequence xn in Y such that

fx2n+1 = Sx2n, fx2n+2 = Tx2n+1, n = 0, 1, 2, ..., and

limn→∞

d(fxn, fxn+1) = 0.

Now we shall show that fxn is Cauchy sequence. Suppose fxn is notCauchy. Then there exists µ > 0 and increasing sequences mk and nk ofpositive integers, such that for all n ≤ mk < nk,

d(fxmk, fxnk

) ≥ µ and d(fxmk, fxnk−1) < µ.

By the triangle inequality,

d(fxmk, fxnk

) ≤ d(fxmk, fxnk−1) + d(fxnk−1, fxnk

).


Making k → ∞, we get

d(fxmk, fxnk

) < µ.

Thus

d(fxmk, fxnk

) → µ as k → ∞.

By (D2) we have

d(fxmk+1, fxnk+1) = d(Sxmk, Txnk

)

≤ ϕ(h(xmk, xnk

))

= ϕ(d(fxmk, fxnk

) + γ[d(Sxmk, fxmk

) + d(Txnk, fxnk

)]).

Making k → ∞

µ ≤ ϕ(µ) < µ, a contradiction.

Thus fxn is Cauchy sequence. Suppose f(Y ) is a complete subspace ofX . Then yn being contained in f(Y ) has a limit in f(Y ). Call it z. Letu = f−1z. Thus fu = z for some u ∈ Y . Note that the subsequences fx2n+1and fx2n+2 also converge to z. Now by (D2),

d(Su, T2n+1) ≤ ϕ(d(fu, f2n+1) + γ[d(Su, fu) + d(T2n+1, f2n+1)]).

Making n → ∞,

d(Su, fu) ≤ ϕ(γd(Su, fu)) < d(Su, fu) a contradiction.

Therefore Su = fu = z. This proves (I). Since S(Y )∪T (Y ) ⊆ f(Y ). Thereforethere exists v ∈ Y such that Su = fv. We claim that fv = Tv. Using (D2),

d(fv, T v) = d(Su, T v)

≤ ϕ(d(fu, fv) + γ[d(Su, fu) + d(Tv, fv)])

= ϕ(γd(fv, T v)) < d(fv, T v),

which is a contradiction. Therefore Tv = fv = Su = fu. This proves (II).Now if Y = X , (S, f) and (T, f) are (IT)-commuting then Sfu = fSu andSSu = Sfu = fSu = ffu, Tfv = fTv and TTv = Tfv = fTv = ffv. Inview of (D2), it follows that

d(SSu, Su) = d(SSu, T v)

≤ ϕ(d(fSu, fv) + γ[d(SSu, fSu) + d(Tv, fv)])

= ϕ(γd(SSu, Su)) < d(SSu, Su).

Therefore SSu = Su = fSu, Su is a common fixed point of S and f . Similarly,Tv is a common fixed point of T and f . Since Su = Tv, we conclude that Su isa common fixed point of S, T and f . The proof is similar when S(Y ) or T (Y )are complete subspaces of X since, S(Y ) ∪ T (Y ) ⊆ f(Y ). Uniqueness of thecommon fixed point follows easily.

Acknowledgements. The authors are indebted to the referee and Prof.Salvador Romaguera for their perspicacious comments and suggestions.


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261–263.[8] G. Jungck and B. E. Rhoades, Fixed points for set-valued functions without continuity,

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East. Asian Math. J. 15 (1999), 211–222.[10] T. C. Lim, On characterization of Meir-Keeler contractive maps, Nonlinear Anal. 46

(2001), 113–120.[11] J. Matkowski, Fixed point theorems for contractive mappings in metric spaces, Cas.

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contractions, Math. Nachr. 127 (1986), 177–180.[14] S. Park and B. E. Rhoades, Meir-Keeler type contractive conditions, Math. Japon. 26

(1981), 13–20.[15] P. D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 64 (2006), 546–

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(1994), 436–440.[17] B. E. Rhoades, A comparison of various definitions of contracting mappings, Trans.

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multivalued mappings, Internat. J. Math. Math. Sci. 7, no. 3 (1984), 429–434.[19] S. Romaguera, Fixed point theorems for mappings in complete quasi-metric spaces, An.

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Received August 2008


S. L. Singh ([email protected])21, Govind Nagar Rishikesh 249201, India

Apichai Hematulin

Department of Mathematics, Nakhonratchasima Rajabhat University, Nakho-ratchasima, Thailand

Rajendra Pant ([email protected])SRM University Modinagar, Ghaziabad (U.P.) 201204, India



Volume 10, No. 1, 2009

pp. 131-157

Well-posedness, bornologies, and the structureof metric spaces

Gerald Beer and Manuel Segura∗

Abstract. Given a continuous nonnegative functional λ that makessense defined on an arbitrary metric space 〈X, d〉, one may considerthose spaces in which each sequence 〈xn〉 for which limn→∞λ(xn) = 0clusters. The compact metric spaces, the complete metric spaces, thecofinally complete metric spaces, and the UC-spaces all arise in this way.Starting with a general continuous nonnegative functional λ defined on〈X, d〉, we study the bornology Bλ of all subsets A of X on whichlimn→∞λ(an) = 0 ⇒ 〈an〉 clusters, treating the possibility X ∈ Bλ asa special case. We characterize those bornologies that can be expressedas Bλ for some λ, as well as those that can be so induced by a uniformlycontinuous λ.

2000 AMS Classification: Primary 54C50, 49K40; Secondary 46A17, 54B20,54E35, 54A10

Keywords: well-posed problem, bornology, UC-space, cofinally complete space,strong uniform continuity, bornological convergence, shielded from closed sets

1. Introduction

In a first course in analysis, one is introduced to two important classes ofmetric spaces as those in which certain sequences have cluster points: a metricspace 〈X, d〉 is called compact if each sequence 〈xn〉 in X has a cluster point,whereas 〈X, d〉 is called complete if each Cauchy sequence in X has a clusterpoint. A Cauchy sequence is one of course for which there exists for each ε > 0a residual set of indices whose terms are pairwise ε-close. If we replace residualby cofinal in the definition, we get a so-called cofinally Cauchy sequence andthe metric spaces X in which each cofinally Cauchy sequence has a cluster pointare called cofinally complete [10, 13, 18, 24, 36]. These are a well-studied classof spaces lying between the compact spaces and the complete ones. Notably,

∗This research was supported by the following grant: NIH MARC U*STAR GM08228.

132 G. Beer and M. Segura

these are the metric spaces that are uniformly paracompact [13, 23, 24, 35]and also those on which each continuous function with values in a metric spaceis uniformly locally bounded [10]. Lying between the compact spaces and thecofinally complete spaces is the class of UC-spaces, also known as Atsuji spaces,which are those metric spaces on which each continuous function with values ina metric space is uniformly continuous [1, 5, 6, 7, 8, 27, 34, 38]. These are alsocalled the Lebesgue spaces [32], as they are those metric spaces 〈X, d〉 for whicheach open cover has a Lebesgue number [1, 8]. These UC-spaces, too, can becharacterized sequentially, as observed by Toader [37]: 〈X, d〉 is a UC spaceif and only if each pseudo-Cauchy sequence in X with distinct terms clusters,where 〈xn〉 is called pseudo-Cauchy [8, p. 59] if for each ε > 0 and n ∈ N, thereexists k > j > n with d(xj , xk) < ε.

It seems worthwhile to study in some organized way classes of metric spaceson which prescribed sequences have cluster points. One program could be tolook at other modifications of the definition of Cauchy sequence, but this ap-proach is limited in scope and is not our purpose here. Instead, given some con-tinuous nonnegative extended real-valued functional λ that makes sense definedon an arbitrary metric space, we look at the class of ”λ-spaces”, i.e., the class ofmetric spaces 〈X, d〉 such that each sequence 〈xn〉 in X with limn→∞λ(xn) = 0has a cluster point. In terms of the language of optimization theory, a spaceis in this class if either infλ(x) : x ∈ X > 0 or the functional λ is Tychonoffwell-posed in the generalized sense [20, 31]. All of the classes mentioned inthe first paragraph fall within this framework. For the compact spaces, thezero functional does the job. For the UC-spaces, the measure of isolation func-tional I(x) = d(x, X\x) is characteristic [1, 8, 27]. For the cofinally completespaces, it is the measure of local compactness functional [10, 13] defined by

ν(x) =

supα > 0 : cl(Sα(x)) is compact if x is a point of local compactness

0 otherwise.

For the complete metric spaces, and paralleling the cofinally complete spacesas we will see in Section 5 infra, it is it is the measure of local completenessfunctional defined by

β(x) =

supα > 0 : cl(Sα(x)) is complete if x has a complete neighborhood

0 otherwise.

In each case discussed above, unless identically equal to ∞, the functional isuniformly continuous; but we do not restrict ourselves in this way, nor do weinsist that our metric spaces be complete.

We find it advantageous to first study more primitively the ”λ-subsets” ofan arbitrary metric space 〈X, d〉: those nonempty subsets A such that eachsequence 〈an〉 within satisfying λ(an) → 0 clusters. In general these form abornology with closed base. As a major result, we characterize those bornolo-gies that arise in this way.

Well-posedness, bornologies, and the structure of metric spaces 133

2. Preliminaries

All metric spaces are assumed to contain at least two points. We denote theclosure, set of limit points and interior of a subset A of a metric space 〈X, d〉by cl(A), A′ and int(A), respectively. We denote the power set of A by P(A)and the nonempty subsets of A by P0(A). We denote the set of all closed andnonempty subsets of X by C0(X), and the set of all closed subsets by C(X).We call A ∈ P0(X) uniformly discrete if ∃ε > 0 such that whenever a1, a2 arein A and a1 6= a2, then d(a1, a2) ≥ ε. If 〈Y, ρ〉 is a second metric space, wedenote the continuous functions from X to Y by C(X, Y ).

If x0 ∈ X and ε > 0, we write Sε(x0) for the open ε-ball with center x0.If A is a nonempty subset of X , we write d(x0, A) for the distance from x0 toA, and if A = ∅ we agree that d(x0, A) = ∞. With d(x, A) now defined, wedenote for ε > 0 the ε-enlargement of A ∈ P(X) by Sε(A), i.e.,

Sε(A) := x ∈ X : d(x, A) < ε =⋃

x∈A

Sε(x).

If A ∈ P0(X) and B ∈ P(X), we define the gap between them by

Dd(A, B) := inf d(a, B) : a ∈ A.

We can define the Hausdorff distance [8, 28] between two nonempty subsetsA and B in terms of enlargements:

Hd(A, B) := inf ε > 0 : A ⊆ Sε(B) and B ⊆ Sε(A).

Hausdorff distance so defined is an extended real-valued pseudometric on P0(X)which when restricted to the nonempty bounded sets is finite valued, and whichwhen restricted to C0(X) is an extended real-valued metric. Hausdorff distancerestricted to C0(X) preserves the following properties of the underlying space(see, e.g., [8, Thm 3.2.4]).

Proposition 2.1. Let 〈X, d〉 be a metric space. The following are true:

(1) 〈C0(X), Hd〉 is complete if and only if 〈X, d〉 is complete;(2) 〈C0(X), Hd〉 is totally bounded if and only if 〈X, d〉 is totally bounded;(3) 〈C0(X), Hd〉 is compact if and only if 〈X, d〉 is compact.

A weaker form of convergence for sequences of closed sets than convergencewith respect to Hausdorff distance is Kuratowski convergence [8, 29, 28]. Givena sequence 〈An〉 in C0(X), we define Li An := x ∈ X : ∀ε > 0, Sε(x) ∩ An 6=∅ residually and Ls An := x ∈ X : ∀ε > 0, Sε(x) ∩ An 6= ∅ cofinally.We say 〈An〉 is Kuratowski convergent to A and write K-limAn = A if A =Li An = Ls An.

The following facts are well-known.


Proposition 2.2. Let 〈An〉 be a sequence in C0(X). Then the following aretrue:

(1) Li An and Ls An are both closed (but perhaps empty);(2) Li An ⊆ Ls An;(3) If An = an, then Li An = lim an if lim an exists and Li An = ∅

if not;(4) If An = an, then Ls An = x : x is a cluster point of 〈an〉 =

⋂∞n∈N

cl(ak : k ≥ n);(5) limHd(An, A) = 0 ⇒ K-limAn = A;(6) If 〈An〉 is decreasing, then K-limAn =

⋂∞n=1 An.

We can also define the Hausdorff measure of noncompactness [4] of a nonemptysubset A in terms of enlargements:

α(A) = inf ε > 0 : A ⊆ Sε(F ), where F is a nonempty finite subset of X.

Clearly, α(A) = ∞ if and only if A is unbounded. The functional α behaves asfollows:

(1) If A ⊆ B, then α(A) ≤ α(B);

(2) α(cl(A)) = α(A);

(3) α(A) = 0 if and only if A is totally bounded;

(4) α(A ∪ B) = max α(A), α(B);

(5) If limHd(An, A) = 0 then limα(An) = α(A).

A famous theorem concerning the Hausdorff measure of noncompactness isKuratowski’s Theorem [4, 29], proved in a novel way below; but first, we stateand prove a useful lemma:

Lemma 2.3. Let 〈X, d〉 be a metric space. Suppose 〈An〉 is a decreasing se-quence in C0(X) which is not Hd-Cauchy. Then ∃ n1 < n2 < n3 < · · · andxnk

∈ Anksuch that xnk

: k ∈ N is uniformly discrete.

Proof. Let 〈An〉 be a decreasing sequence in C0(X) that is not Hd-Cauchy.Then ∃ε > 0 such that ∀n0 ∈ N, ∃m > n > n0 such that Hd(Am, An) > ε.Choose m1, m2 with m2 > m1 > 1 such that Hd(Am1

, Am2) > ε. Then let

i1 > m2, and choose m3, m4 with m4 > m3 > i1 such that Hd(Am3, Am4

) >ε. Then let i2 > m4, and choose m5, m6 with m6 > m5 > i2 such thatHd(Am5

, Am6) > ε. Continuing, we have m1 < m2 < m3 < · · · such that

Hd(Am2j−1, Am2j

) > ε where j > 1. Now for i > 1, pick xni∈ Am2i−1

withd(xni

, Am2i) > ε. Then for i < k,

d(xni, xnk

) > d(xni, Am2k−1

) > d(xni, Am2k−2

) > d(xni, Am2i

) > ε.

We have shown 〈xni〉 has distinct terms and is a uniformly discrete sequence.


Here is our novel proof (in one direction) of Kuratowski’s Theorem based oncompleteness of 〈C0(X), Hd〉.

Theorem 2.4 (Kuratowski’s Theorem on Completeness). Let 〈X, d〉 bea metric space. Then 〈X, d〉 is complete if and only if whenever 〈An〉 is adecreasing sequence in C0(X) with lim α(An) = 0, then A :=

⋂

n∈NAn 6= ∅.

Proof. Suppose 〈X, d〉 is complete and 〈An〉 is a decreasing sequence in C0(X)with limα(An) = 0. Suppose 〈An〉 is not Hd-Cauchy. Then ∃n1 < n2 < n3 < ...and xnk

∈ Anksuch that 〈xnk

〉 is a uniformly discrete sequence with distinctterms. Then ∃ε > 0 such that d(xni

, xnj) > ε where i 6= j. Hence ∀n ∈

N, α(An) ≥ ε2 ⇒ limα(An) ≥ ε

2 , which is a contradiction. Thus 〈An〉 mustbe Hd-Cauchy. Since 〈X, d〉 is complete, by Proposition 2.1(1) 〈C0(X), Hd〉 iscomplete, so ∃B ∈ C0(X) with B = Hd − lim〈An〉. Since limHd(An, B) = 0,by Proposition 2.2(5) B = K − limAn, . Since 〈An〉 is decreasing in C0(X), byProposition 2.2(6) K-lim An =

⋂∞n=1 An. Hence B = A ⇒ A 6= ∅.

Conversely, suppose whenever 〈An〉 is a decreasing sequence in C0(X) withlimα(An) = 0, then A 6= ∅. Then if 〈xn〉 is a Cauchy sequence, we have⋂∞

n∈Nclxk : k ≥ n nonempty, so by Proposition 2.2(4), 〈xk〉 has a cluster

point. Hence 〈X, d〉 is complete.

Let f : X → [0,∞]. Then by saying f is lower semi-continuous at a pointx0 ∈ X , we mean whenever α < f(x0), α ∈ R, then ∃δ > 0 such that ∀x ∈Sδ(x0), f(x) > α. By saying f is upper semi-continuous at a point x0 ∈ X , wemean whenever α > f(x0), α ∈ R, then ∃δ > 0 such that ∀x ∈ Sδ(x0), f(x) <α. Note that f(x0) = ∞ ⇒ f is upper semi-continuous at x0, and f(x0) = 0 ⇒f is lower semi-continuous at x0. If f is both upper and lower semi-continuousat a point x0 ∈ X , then we say f is continuous at x0.

3. Notes on nonnegative continuous functionals

We first discuss a framework in which many important nonnegative contin-uous functionals arise.

Let P be an hereditary property of open subsets of 〈X, d〉: if V, W are opensets where W ⊆ V then P (V ) ⇒ P (W ). Define λP : X → [0,∞] by

λP (x) =

supα > 0 : P (Sα(x)) if ∃α > 0 where P (Sα(x));

0 otherwise.

Example 3.1. Consider P (V ) := V contains at most one point. Then theresulting λP is the measure of isolation functional λP (x) = I(x) := d(x, X \x).

Example 3.2. Next consider P (V ) := V ∩ E = ∅ where E ⊆ X . Then theinduced λP gives the distance from a variable point of X to the set E.

Example 3.3. Now consider the case when P (V ) := cl(V ) is compact. Thenthe resulting λP is the measure of local compactness functional ν giving thesupremum of the radii of the closed balls with center x that are compact.


Example 3.4. A final example of an hereditary property P is P (V ) := V iscountable.

Of particular importance is the kernel of the metric space 〈X, d〉 with re-spect to a continuous function λ : X → [0,∞], which we define as Ker(λ) :=x ∈ X : λ(x) = 0. If we consider the resulting λP from Example 3.1, thenKer(λP )= X ′, the set of limit points of X . For λP from Example 3.2, weget Ker(λP )= cl(E). For λP from Example 3.3, Ker(λP ) equals the points ofnon-local compactness of X . Finally, if we consider the corresponding λP forExample 3.4, then Ker(λP ) equals the set of condensation points of X .

Proposition 3.5. Let P be an hereditary property of open sets in 〈X, d〉. IfλP (x0) = ∞ for some x0 ∈ X, then λP (x) = ∞ for all x ∈ X. Otherwise ifλP is finite valued, then λP is 1-Lipschitz.

Proof. Suppose λP (x0) = ∞ for some x0 ∈ X . Let x ∈ X where x0 6= x,and let α > 0 be arbitrary. Since supµ > 0 : P (Sµ(x0)) = ∞, ∃α0 > 0such that P (Sα0

(x0)) and Sα(x) ⊆ Sα0(x0), so that P (Sα(x)). This shows

that λP (x) = ∞ for all x ∈ X . Otherwise, λP is finite valued. If λP failsto be 1-Lipschitz, there exist x, w ∈ X with λP (x) > λP (w) + d(x, w). Takean α > 0 where λP (x) > α > λP (w) + d(x, w), so that P (Sα(x)). ThenSα−d(x,w)(w) ⊆ Sα(x), so P (Sα−d(x,w)(w)). However, α − d(x, w) > λP (w),which is a contradiction. Hence, λP is 1-Lipschitz.

We next introduce the induced set functional λ : P0(X) → [0,∞] that we willuse to characterize λ-spaces in Section 5:

λ(A) := supλ(a) : a ∈ A,

where λ : X → [0,∞] is a continuous functional. The following proposition

lists obvious properties of the set functional λ.

Proposition 3.6. Let 〈X, d〉 be a metric space and let λ : P0(X) → [0,∞] beas defined above. Then the following are true for nonempty subsets A, B:

(1) λ(A ∪ B) = maxλ(A), λ(B);

(2) λ(cl(A)) = λ(A);

(3) λ(A) = 0 if and only if A ⊆ Ker(λ).

It is now useful to introduce a strengthening of uniform continuity of afunction restricted to a subset of X as considered in [14, 15].

Definition 3.7. Let 〈X, d〉 and 〈Y, ρ〉 be metric spaces and let A be a subsetof X. We say that a function f : X → Y is strongly uniformly continuous onA if ∀ε > 0 ∃δ > 0 such that if d(x, w) < δ and x, w ∩ A 6= ∅, thenρ(f(x), f(w)) < ε.

Note that strong uniform continuity on A = x0 means simply that f iscontinuous at x0. Strong uniform continuity on A = X is uniform continuity. Acontinuous function on X is strongly uniformly continuous on each nonempty


compact subset, not merely uniformly continuous when restricted to such asubset.

Lemma 3.8. Let λ : X → [0,∞] be continuous. If λ is finite-valued and

strongly uniformly continuous on A ∈ P0(X) then λ is Hd-continuous at A.

Proof. We show that λ is lower and upper semi-continuous at A, respectively.For lower semi-continuity, we have nothing to show if λ(A) = 0. Otherwise,

fix α0 > 0 and suppose α0 < λ(A). Then ∃a0 ∈ A such that λ(a0) > α0 + ε0,where ε0 > 0. Choose by strong uniform continuity of λ on A δ0 > 0 suchthat if a ∈ A, x ∈ X and d(a, x) < δ0, then |λ(a) − λ(x)| < ε0. Now supposeHd(A, B) < δ0; choose b ∈ B such that d(a0, b) < δ0. Then |λ(a0) − λ(b)| <ε0 ⇒ λ(b) > α0 ⇒ λ(B) > α0.

For upper semi-continuity, we have nothing to show if λ(A) = ∞. Otherwise,

fix α1 > 0 with λ(A) < α1. Fix ε1 > 0 so that ∀a ∈ A, λ(a) < α1 − ε1

2 . Letδ1 > 0 be such that if a ∈ A, x ∈ X and d(a, x) < δ1 then |λ(a) − λ(x)| < ε1

3 .Suppose Hd(A, B) < δ1 and let b ∈ B be arbitrary. Choose a ∈ A such thatd(a, b) < δ1. Then

|λ(a) − λ(b)| < ε1

3 ⇒ −ε1

3 < λ(a) − λ(b) < α1 −ε1

2 − λ(b)

⇒ λ(b) < α1 −ε1

6 .

Since b ∈ B was arbitrary, λ(B) < α1.

The next counterexample shows that when λ is not strongly uniformly con-tinuous on A, it is not guaranteed that the λ functional is Hd-continuous atA.

Example 3.9. Let X = [0,∞)× [0,∞) and define λ : X → [0,∞) by λ(x, y) =xy. Let A = (0, y) : y ∈ [0,∞). Obviously, λ is not strongly uniformlycontinuous on A, since one can take ε = 1 and for any δ > 0, if n > 2

δ we have

d((0, n), ( 2n , n)) < δ, but |λ(0, n)−λ( 2

n , n)| = 2 > ε. If we let An = 1n×[0,∞),

then 〈An〉Hd→ A. But for all n, λ(An) = ∞ while λ(A) = 0, showing λ is not

Hd-continuous at A.

4. λ-Subsets

Definition 4.1. Let 〈X, d〉 be a metric space, and λ : X → [0,∞] be contin-uous. We say A ∈ P0(X) is a λ-subset of X if whenever 〈an〉 is a sequencein A and λ(an) → 0, then 〈an〉 has a cluster point in X. When X is itself aλ-subset, then 〈X, d〉 is called a λ-space.

We denote the family of λ-subsets by Bλ. Note that Bλ is not altered byreplacing λ by minλ, 1, if one is bothered by functionals that naturallyassume values of ∞. We now provide some examples.

Example 4.2. If 〈X, d〉 is any metric space, and λ(x) ≡ 1, then Bλ = P0(X).

Example 4.3. If 〈X, d〉 is a metric space, then the family of nonempty subsetswith compact closure K0(X) is Bλ for the zero functional λ on X .


Example 4.4. If 〈X, d〉 is an unbounded metric space and x0 ∈ X , thenthe family of nonempty d-bounded subsets Bd(X) is Bλ for the continuousfunctional on X defined by

λ(x) =1

1 + d(x, x0)

Notice here that while inf λ(X) = 0, we have Ker(λ) = ∅. We shall seepresently that Bλ for all such λ-functionals arises in this way (see Theorem4.17 infra).

Example 4.5. The λ-subsets of a metric space corresponding to the measureof isolation functional I(x) = d(x, X\x) are called the UC-subsets, as studiedin [15]. The λ-subsets of a metric space corresponding to the measure of localcompactness functional ν are called the cofinally complete subsets, as studiedin [13].

Definition 4.6. Let X be a topological space. We call a family of nonemptysubsets A of X a bornology [9, 14, 22, 30] provided

(1)⋃

A = X;(2) A1, A2, A3, ..., An ⊆ A ⇒

⋃ni=1 Ai ∈ A;

(3) A ∈ A and ∅ 6= B ⊆ A ⇒ B ∈ A.

We will of course be focusing on bornologies in a metric space 〈X, d〉. Thelargest bornology is P0(X) and the smallest is the set of nonempty finite subsetsF0(X). The bornologies K0(X) and Bd(X) lie between these extremes. Ofimportance in the sequel are functional bornologies, that is, bornologies arisingas the family of subsets on which a real-valued function with domain X isbounded. The proof of the next proposition is left to the reader, and it impliesthat K0(X) is the smallest possible Bλ.

Proposition 4.7. Let λ : X → [0,∞] be continuous. Then Bλ forms a bornol-ogy containing the nonempty compact subsets.

By a base for a bornology, we mean a subfamily that is cofinal in the bornol-ogy with respect to inclusion. For example, a countable base for the metricallybounded subsets of 〈X, d〉 consists of all balls with a fixed center and integralradius. The next result says that Bλ has a closed base, that is, a base thatconsists of closed sets.

Proposition 4.8. Let λ : X → [0,∞] be continuous, and let A be a λ-set.Then cl(A) is also a λ-set.

Proof. Let 〈xn〉 be a sequence in cl(A) where λ(xn) → 0. We may assume∀n ∈ N that λ(xn) < ∞. By the continuity of λ, ∃ a sequence 〈an〉 in A where∀n ∈ N, d(xn, an) < 1

n and λ(an) < λ(xn) + 1n . Then since 〈an〉 has a cluster

point, 〈xn〉 must have one also.


The following elementary proposition was not noticed for either the bornol-ogy of UC-subsets or the bornology of cofinally complete subsets. It will beused to characterize those bornologies that are Bλ for some λ ∈ C(X, [0,∞)).

Proposition 4.9. Let 〈X, d〉 be a metric space and let λ : X → [0,∞] becontinuous. Suppose B is a nonempty closed subset of X. Then B is a λ-subset if and only if B ∩ Ker(λ) is compact, and whenever A is a nonemptyclosed subset of B with A ∩ Ker(λ) = ∅, then inf λ(A) > 0.

Proof. Suppose first that B is a λ-set. Then each sequence in B ∩ Ker(λ) is aminimizing sequence and since B ∩Ker(λ) is closed, the sequence clusters to apoint of B ∩Ker(λ). Suppose next that A ∈ C0(X)∩ P0(B) does not intersectKer(λ), yet inf λ(A) = 0. Then λ has a minimizing sequence in A that clustersto a point of A which by continuity also must be in Ker(λ) , contradictingA ∩ Ker(λ) = ∅.

Conversely, suppose B satisfies the two conditions, and 〈bn〉 is a sequence inB with λ(bn) → 0 but that does not cluster. By the assumed compactness ofB ∩ Ker(λ), and by passing to a subsequence, we may assume that ∀n, bn /∈Ker(λ). But then with A = bn : n ∈ N, the second condition is violated.

Proposition 4.10. Let λ : X → [0,∞] be continuous. Then a λ-set A iscompact if and only if ∀ε > 0, Bε := a ∈ A : λ(a) ≥ ε is compact.

Proof. Let A be compact λ-set. Since λ is a continuous function, x : λ(x) ≥ εis closed. Since Bε = A ∩ x : λ(x) ≥ ε, Bε is compact.

Conversely, suppose 〈an〉 is an arbitrary sequence in A. If λ(an) → 0, thenthe sequence clusters because A is a λ-set. Otherwise, ∃ε > 0 and an infinitesubset N1 of N such that ∀n ∈ N1, λ(an) ≥ ε. Hence 〈an〉n∈N1

is a sequence inthe compact set Bε. Thus, the sequence 〈an〉 clusters, and A is compact.

Our next proposition involves λ-subsets and strong uniform continuity.

Proposition 4.11. Let λ : X → [0,∞) be continuous.

(1) If A is a λ-subset, λ is strongly uniformly continuous on A, and 〈xn〉is a sequence in X with lim d(xn, A) = 0 and lim λ(xn) = 0, then 〈xn〉clusters.

(2) Strong uniform continuity of λ on each member of Bλ coincides withglobal uniform continuity.

Proof. We prove statement (2), leaving (1) to the reader. Suppose λ fails to beglobally uniformly continuous. Then for some ε > 0, there exist sequences 〈xn〉and 〈wn〉 in X such that for each n, d(xn, wn) < 1

n yet f(xn) + ε < f(wn).While B := wn : n ∈ N is in Bλ, λ is not strongly uniformly continuous onB.

Example 4.12. For a counterexample to Proposition 4.11(1), let us revisitthe metric space X and the functional λ of Example 3.9. Then A := (x, y) :xy = 1 ∪ (x, y) : x = y and x ≤ 1, as shown in Figure 1, is a λ-set. If


xn = (0, n), then lim d(xn, A) = 0 and limλ(xn) = 0, but the sequence 〈xn〉does not cluster.

Figure 1

The next result is anticipated by a decomposition theorem for spaces onwhich a continuous function that is Tychonoff well-posed in the generalizedsense is defined [31, Prop 10.1.7]. It is also anticipated by particular decom-position theorems in the special cases of the bornology of UC-subsets and thebornology of cofinally complete subsets [13, 15] (see previously for UC spacesand cofinally complete spaces [8, 10, 23]).

Theorem 4.13. Let 〈X, d〉 be a metric space and suppose λ ∈ C(X, [0,∞]).Then A ∈ P0(X) is a λ-subset if and only if cl(A) ∩ Ker(λ) is compact and∀δ > 0, ∃ε > 0 such that a ∈ A \ Sδ(cl(A) ∩ Ker(λ)) ⇒ λ(a) > ε.

Proof. First, suppose cl(A) ∩ Ker(λ) is not compact, and therefore nonempty.Choose a sequence 〈an〉 in cl(A)∩Ker(λ) with no cluster point. Then λ(an) →0, but 〈an〉 has no cluster point ⇒ cl(A) is not a λ-set ⇒ A is not a λ-set.Suppose now that for some δ > 0 that infλ(a) : a ∈ A \Sδ(cl(A)∩Ker(λ)) =0. Select an ∈ A \ Sδ(cl(A) ∩ Ker(λ)) with λ(an) < 1

n . There can be nopossible cluster point p for 〈an〉 as by continuity λ(p) = 0 must hold, whiled(p, cl(A) ∩ Ker(λ)) ≥ δ. Again, A is not a λ-set.

Conversely, suppose cl(A) ∩ Ker(λ) is compact, and ∀δ > 0, ∃εδ > 0 suchthat a ∈ A \ Sδ(cl(A) ∩ Ker(λ)) ⇒ λ(a) > εδ. Let 〈an〉 be a sequence in Awhere λ(an) → 0. If cl(A) ∩ Ker(λ) = ∅, then A = A \ Sδ(cl(A) ∩ Ker(λ))for each δ. So then given δ > 0, ∀n λ(an) > εδ, which is a contradiction. Weconclude that cl(A) ∩ Ker(λ) 6= ∅. Then given δ > 0, λ(an) ≤ εδ eventually⇒ an ∈ Sδ(cl(A) ∩ Ker(λ)) eventually ⇒ 〈an〉 has a cluster point by thecompactness of cl(A) ∩ Ker(λ).

We now address a basic question: what are necessary and sufficient condi-tions on a bornology B in a metric space 〈X, d〉 such that B = Bλ for some


λ ∈ C(X, [0,∞))? The key tools in answering this question are Proposition 4.9and an important lemma of S.-T. Hu ([25, Thm 13.2] or [26, p. 189]), provedusing the Urysohn Lemma.

Lemma 4.14 (Hu’s Lemma). Let B 6= P0(X) be a bornology on a normaltopological space X having a countable base Bn : n ∈ N such that ∀n ∈N, cl(Bn) ⊆ int(Bn+1). Then there exists an unbounded f ∈ C(X, [0,∞)) suchthat

B = A : f(A) is a bounded set of reals.

It is easy to see that the conditions of the lemma are satisfied if and only if(1) ∀B ∈ B, B 6= X ; (2) B has a countable base; (3) B has an open base; and(4) B has a closed base.

To obtain our characterization, we break our λ-functionals into two classes:those for which Ker(λ) = ∅, and those for which Ker(λ) 6= ∅. We need animmediate consequence of Theorem 4.13 to deal with the first situation thatwe record as a lemma.

Lemma 4.15. Let λ ∈ C(X, [0,∞)) have no minimum value, yet inf λ(X) = 0.Then Bλ = A ∈ P0(X) : inf λ(A) > 0.

Theorem 4.16. Let B be a bornology on 〈X, d〉. Then B = Bλ for someλ ∈ C(X, [0,∞)) with Ker(λ) = ∅ if and only if B has a countable base Bn :n ∈ N such that ∀n ∈ N, cl(Bn) ⊆ int(Bn+1).

Proof. For sufficiency, if X ∈ B, we can put λ(x) ≡ 1. Otherwise, applying Hu’sLemma to generate an unbounded f ∈ C(X, [0,∞)), put λ(x) := (1 + f(x))−1.Noting that λ is bounded away from zero on a subset of X if and only if f isbounded above on the subset, we see by Lemma 4.15 that λ does the job.

For necessity, if B = Bλ where Ker(λ) = ∅, then by Lemma 4.15, B ∈ B ⇔inf λ(B) > 0. By the continuity of λ, λ−1([ 1

n ,∞)) : n ∈ N is the desiredcountable base.

Theorem 4.17. Let B be a bornology on 〈X, d〉. The following conditions areequivalent:

(1) B = Bλ for some λ ∈ C(X, [0,∞)) with Ker(λ) = ∅;(2) B = Bρ(X) for some metric ρ equivalent to d.

Proof. (2) ⇒ (1). If ρ is a bounded metric, take λ(x) ≡ 1. Otherwise, we invokeTheorem 4.16 for Bρ(X), putting Bn := x : ρ(x, x0) ≤ n where x0 ∈ X isfixed.

(1) ⇒ (2). If Bλ = P0(X), take ρ = min1, d. Otherwise, with Bn =λ−1([ 1

n ,∞)) 6= X , apply Hu’s Lemma to once again generate an unbounded f .The metric

ρ(x, w) := min1, d(x, w) + |f(x) − f(w)|

satisfies Bλ = Bρ(X) and is equivalent to d.


We now come to the harder part.


(1) B = Bλ for some λ ∈ C(X, [0,∞)) with Ker(λ) 6= ∅;(2) B has a closed base, and ∃C ∈ C0(X) with open neighborhoods Vn : n ∈

N satisfying ∩∞n=1Vn = C and ∀n ∈ N, cl(Vn+1) ⊆ Vn such that ∀B ∈

C0(X), B ∈ B ⇔ B ∩ C is compact, and whenever A is a nonemptyclosed subset of B disjoint from C, then for some n, A ∩ Vn = ∅.

Proof. (1) ⇒ (2). By Proposition 4.8, B has a closed base, and by Proposition4.9, we can take C = Ker(λ) and Vn = λ−1([0, 1

n )).

(2) ⇒ (1). We consider several cases for the set C. First if C = X , thena nonempty closed set B is in B if and only if B is compact, and since thebornology has a closed base, it is the bornology K0(X) of nonempty subsetswith compact closure and with λ(x) ≡ 0, we get B = Bλ. A second possibilityis that C = Vn ⊂ X for some n. Since C, X\C forms a nontrivial separationof X , the function λ assigning 0 to each point of C and 1 to each point of X\Cis continuous. We intend to show that B = Bλ.

Since both bornologies have closed bases, it suffices to show closed membersof one belong to the other. If B ∈ B ∩ C0(X), then any minimizing sequencein B lies eventually in C, and since B ∩ C is compact, it clusters. This showsB ∈ Bλ. For the reverse inclusion, if B ∈ Bλ is closed, then B ∩ Ker(λ) iscompact, that is, B ∩ C is compact. Also if A is a closed subset of B disjointfrom C, then A ∩ Vn = ∅ without any consideration of λ.

In the remaining case we may assume without loss of generality that ∀n ∈N, C ⊂ Vn ⊂ X . We now apply Hu’s Lemma to the metric subspace X\C withrespect to the bornology having the closed base X\Vn : n ∈ N. We producean unbounded continuous f : X\C → [0,∞) such that ∀A ∈ P0(X\C), f(A)is bounded if and only if for some n, A ⊆ X\Vn. We next define our functionλ by

λ(x) =

0 if x ∈ C1

1+f(x) otherwise.

Evidently λ is continuous restricted to the open set X\C. Given ε ∈ (0, 1),choose n ∈ N with x ∈ X\C : f(x) ≤ 1−ε

ε ⊆ X\Vn. It follows that ∀x ∈ Vn,we have λ(x) < ε, establishing global continuity of λ.

Again we must show that B ∩ C0(X) = Bλ ∩ C0(X). For a closed set B,B ∩Ker(λ) is compact if and only if B ∩C is compact because by constructionKer(λ) = C. If B ∈ C0(X) and A is a nonempty closed subset with A ∩ C =A ∩ Ker(λ) = ∅ then ∃n with A ∩ Vn = ∅ ⇔ ∃n with A ⊆ X\Vn ⇔ f isbounded above on A ⇔ inf λ(A) > 0. The result now follows from Proposition4.9.


We next show that that there are bornologies with closed base that fail tobe a bornology of λ-subsets.

Example 4.19. Consider R with the zero-one metric and and let B be thebornology of countable nonempty subsets. Since R is uncountable, B fails tohave a countable base. By Theorem 4.16, it remains to show that B cannot beBλ for any λ with nonempty kernel. We show that condition (2) of Theorem4.18 cannot hold. Suppose to the contrary that such a C with neighborhoodsVn : n ∈ N existed. Since the intersection of C with each countable set mustbe compact, we conclude C is finite. For each n, put Bn := X\Vn. Clearly,Bn ∩ C is compact as it is empty. Also each (closed) subset of Bn is triviallydisjoint from Vn. By condition (2) of Theorem 4.18, Bn must be countable,and since X\C = ∪∞

n=1Bn, it too must be countable. This is a contradiction,and so the bornology of countable subsets cannot be a bornology of λ-subsets.

Here is a natural follow-up question: when is a bornology B a bornologyof λ-subsets for some uniformly continuous λ : X → [0,∞)? In our analysis,strong uniform continuity of a function on members of a bornology plays a keyrole. We first obtain an analog of Hu’s Lemma, which is implicit in the proofof [14, Thm. 3.18].

Lemma 4.20. Suppose B is a bornology on a metric space 〈X, d〉 that doesnot contain X. Suppose B has a countable base Bn : n ∈ N such that∀n ∈ N, ∃δn > 0 with Sδn

(Bn) ⊆ Bn+1. Then there exists an unboundedf ∈ C(X, [0,∞)) such that f is strongly uniformly continuous on each Bn andsuch that

B = A : f(A) is a bounded set of reals.

Proof. For each n ∈ N let fn : X → [0, 1] be the uniformly continuous functiondefined by fn(x) = min1, 1

δnd(x, Bn). The values of fn all lie in [0, 1], and

fn(Bn) = 0 and fn(X\Bn+1) = 1. Put f = f1 + f2 + f3 + · · · . First notethat the restriction of f to each Bn agrees with f1 + f2 + f3 + · · · + fn−1 sothat

(1) ∀n, f restricted to Bn is uniformly continuous;

(2) ∀n, f(Bn) ⊆ [0, n− 1].

By (1) f is strongly uniformly continuous on each Bn because f is uniformlycontinuous restricted to Bn+1 and this larger set contains an enlargement ofBn. By (2) ∀n, f(Bn) is bounded, so f restricted to each member of B isbounded because Bn : n ∈ N is a base. Finally, if f(A) is bounded, then forsome n, A ⊆ Bn because x /∈ Bn+1 ⇒ f(x) ≥ n .

We note that the function f in the Lemma 4.20 is strongly uniformly contin-uous on each member of B. More generally, the sets on which a continuous realfunction g is strongly uniformly continuous always form a bornology containingthe UC-subsets; in fact, the UC-subsets form the largest common bornology asg runs over C(X, R) [15]. We also note that if δn can be chosen independent of


n in the statement of Lemma 4.20, one can construct a uniformly continuousfunction f , but the proof is a little more delicate [9, Thm. 4.2].

We will need the following fact about strong uniform continuity.

Proposition 4.21. Suppose g : 〈X, d〉 → (0,∞) is strongly uniformly contin-uous on a nonempty subset A of X, and g is bounded away from zero in someenlargement of A. Then λ(x) := 1

g(x) is strongly uniformly continuous on A.

Proof. Suppose ∀x ∈ Sδ(A), we have g(x) ≥ α > 0. Given ε > 0, ∃δε ∈ (0, δ)such that if a ∈ A and x ∈ X and d(a, x) < δε, then |g(x) − g(a)| < εα2. Wecompute

|λ(x) − λ(a)| =

∣

∣

∣

∣

1

g(x)−

1

g(a)

∣

∣

∣

∣

=|g(a) − g(x)|

|g(x)g(a)|,

and since a, x ⊆ Sδε(A) ⊆ Sδ(A), we further have

|g(a) − g(x)|

|g(x)g(a)|<

εα2

|g(x)g(a)|≤

εα2

α2= ε,

and this yields |λ(x) − λ(a)| < ε.


(1) B = Bλ for some uniformly continuous λ : X → [0,∞) with Ker(λ) =∅;

(2) B has a countable base Bn : n ∈ N such that ∀n ∈ N, ∃δn > 0 withSδn

(Bn) ⊆ Bn+1.

Proof. (1) ⇒ (2). If inf λ(X) > 0, then X ∈ B and we can put Bn := Xfor each n ∈ N. Otherwise, put Bn = λ−1([ 1

n ,∞)) ∈ B; choose by uniformcontinuity of λ a positive δn such that

d(x, w) < δn ⇒ |f(x) − f(w)| <1

n−

1

n + 1.

Then we have ∀n ∈ N, Sδn(Bn) ⊆ Bn+1.

(2) ⇒ (1) The case X ∈ B, that is B = P0(X), is of course trivial. Otherwise,we take f as guaranteed by Lemma 4.20 and as expected put λ(x) = (1 +f(x))−1. We use Proposition 4.11(2) to establish uniform continuity. Fix n ∈ N.We know g(x) := 1 + f(x) is strongly uniformly continuous on Bn and that gis bounded below by 1 on all of X . Taking the reciprocal, by Proposition 4.21,we see that λ is strongly uniformly continuous on each Bn and thus on eachB ∈ B, as required.

As expected, the bornologies that fulfill the conditions of Theorem 4.22 aremetric boundedness structures [9], that is, they are of the form Bρ for certain ρequivalent to d. In turns out that the metrics ρ are those for which the identity


id : 〈X, d〉 → 〈X, ρ〉 is strongly uniformly continuous on each ρ-bounded subset.We leave this as an exercise to the interested reader, following the proof ofTheorem 4.17 (see also [14]).


(1) B = Bλ for some uniformly continuous λ : X → [0,∞) with Ker(λ) 6=∅;

(2) B has a closed base, and ∃C ∈ C0(X) with open neighborhoods Vn : n ∈N satisfying ∩∞

n=1Vn = C and ∀n ∈ N, ∃δn > 0 with Sδn(Vn+1) ⊆ Vn

such that ∀B ∈ C0(X), B ∈ B ⇔ B ∩ C is compact, and wheneverA is a nonempty closed subset of B disjoint from C, then for somen, A ∩ Vn = ∅.

Proof. (1) ⇒ (2). Let λ satisfy condition (1), and put C = Ker(λ). If C =X, ∀n ∈ N, put Vn = X . Otherwise, ∃k ∈ N and x ∈ X with λ(x) > 1

k . In

this case ∀n ∈ N, put Vn := x ∈ X : λ(x) < 1n+k. By uniform continuity of

λ, ∃δn > 0 with

d(x, w) < δn ⇒ |λ(x) − λ(w)| <1

n + k−

1

n + k + 1

which means that Sδn(Vn+1) ⊆ Vn. By Proposition 4.8 and Proposition 4.9,

Bλ satisfies the conditions on a bornology B listed in (2).

(2) ⇒ (1). We handle this implication by modifying the proof of (2) ⇒ (1) inTheorem 4.18. The case C = X is handled in exactly the same manner. In thecase that C = Vn ⊂ X for some n, we define a uniformly continuous functionλ on X by λ(x) = min 1

δnd(x, C), 1. Since Sδn

(C) ⊆ Vn, we see that λ maps

each point of X\C to 1 and each point of C to 0. Verification that B = Bλ

proceeds exactly as in the proof of Theorem 4.18.In the remaining case we can assume for each n ∈ N that C ⊂ Vn ⊂ X . By

condition (2), ∀n ∈ N, we have

Sδn(X\Vn) ⊆ X\Vn+1.

We now apply Lemma 4.20 to the space X\C equipped with the bornologywith base X\Vn : n ∈ N to produce an unbounded f : X\C → [0,∞) that isstrongly uniformly continuous on each set X\Vn and such that f(A) is boundedif and only if A is a subset of some X\Vn. We now define λ : X → [0,∞) by

λ(x) =

0 if x ∈ C1

1+f(x) otherwise.

The proof of Theorem 4.22 shows that the restriction of λ to X\C is uniformlycontinuous, so if λ fails to be globally uniformly continuous, ∃ε > 0 such that


∀k ∈ N, ∃ck ∈ C and xk ∈ X\C such that d(ck, xk) < 1k while λ(xk) > ε. Now

as λ is bounded below by ε on xk : k ∈ N, f is bounded above so restricted.It follows that for some n0 ∈ N, we have xk : k ∈ N∩Vn0

= ∅. But choosing1k < δn0

, by condition (2)

d(ck, xk) <1

k⇒ ck ∈ X\Vn0+1.

This is a contradiction because X\Vn0+1 ∩ C = ∅. This contradiction estab-lishes global uniform continuity, and agreement of the bornologies is argued asbefore.

To end this section, we note that convergence in Hausdorff distance neednot preserve λ-sets, even when the λ-functional is uniformly continuous.

Example 4.24. Let λ : R2 → [0,∞), where λ(x, y) = y, and for each positiveinteger n put An := (x, 0) : x ∈ [0, n] ∪ (x, y) : y = 1

n (x − n), x ∈ [n, n +

1] ∪ (x, y) : y = 1n , x ≥ n + 1, as shown in Figure 2. Then λ is uniformly

continuous and 〈An〉 is a sequence of closed λ-sets converging in Hausdorffdistance to A, where A := (x, y) : y = 0, x ≥ 0. But A is not a λ-set.

Figure 2

5. λ-Spaces

Given a continuous nonnegative function λ on a metric space 〈X, d〉, recallthat X is called a λ-space provided each sequence 〈xn〉 in X with lim λ(xn) = 0has a cluster point. As noted in the introduction, if λ is defined appropriately,the λ-spaces include the compact metric spaces, the UC-spaces and the cofinallycomplete metric spaces. We now show that they include the complete metricspaces.


Proposition 5.1. Let 〈X, d〉 be a metric space, and let P (V ) mean cl(V ) is acomplete subspace equipped with the metric d. Put β := λP , so that

β(x) =

supα > 0 : cl(Sα(x)) is complete if ∃α > 0 with cl(Sα(x)) complete;

0 otherwise.

Then 〈X, d〉 is a complete metric space if and only if 〈X, d〉 is a β-space.

Proof. Proving this is straightforward. First suppose 〈X, d〉 is complete, so∀x ∈ X, β(x) = ∞. Each sequence 〈xn〉 with limβ(xn) = 0 has a cluster pointas this is true vacuously. Hence 〈X, d〉 is a β-space.

To see the converse, suppose 〈X, d〉 is a β-space and 〈xn〉 is a Cauchy se-quence. There are two possibilities: (1) limβ(xn) = 0, and (2) lim sup β(xn) >0. If limβ(xn) = 0, then there exists a cluster point by the definition ofa β-space. Otherwise ∃ε > 0 and and infinite subset N1 of N such that∀n ∈ N1, β(xn) > ε. Choose k ∈ N such that if n > m > k, then d(xn, xm) < ε.If n1 ∈ N1 and n1 > k, then x : d(x, xn1

) ≤ ε contains a tail of 〈xn〉 that isalso Cauchy. Since β(xn1

) > ε, x : d(x, xn1) ≤ ε is complete, which implies

the tail has a cluster point, so 〈xn〉 has a cluster point also. Hence 〈X, d〉 iscomplete.

Proposition 4.9 and Theorem 4.13 provide characterizations of λ-spaces,which we now list.

Theorem 5.2. Let 〈X, d〉 be a metric space, and let λ : X → [0,∞] be contin-uous. The following are equivalent:

(1) 〈X, d〉 is a λ-space;

(2) Ker(λ) is compact, and if A ∈ C0(X) with A ∩ Ker(λ) = ∅, then infλ(A) > 0;

(3) Ker(λ) is compact, and ∀δ > 0, ∃ε > 0 such that d(x,Ker(λ)) > δ ⇒λ(x) > ε.

Although all λ-spaces must have a compact kernel, it is easy to produceexamples showing that this alone is not sufficient (see, e.g., [31, Ex. 10.1.3]).The following proposition shows how normal pathology is in this regard.

Proposition 5.3. Let 〈X, d〉 be a noncompact metric space and let C be anarbitrary compact subset. Then there exists λ ∈ C(X, [0,∞)) with Ker(λ) = Cfor which X is not a λ-space.

Proof. Pick distinct points x1, x2, x3, . . . in X\C such that 〈xn〉 has no clusterpoint. Note that A := xn : n ∈ N is a closed discrete set. If C = ∅, chooseby the Tietze Extension Theorem [21, p. 149] f ∈ C(X, [0,∞)) satisfyingf(xn) = n, and clearly λ(x) = (1+f(x))−1 does the job. When C is nonempty,by the Tietze Extension Theorem, there is a nonnegative continuous functionλ1 on X mapping C to 0 such that ∀n, λ1(xn) = 1

n . The desired λ is definedby λ(x) = λ1(x) + d(x, A ∪ C).


The last result of course shows that whenever C is a nonempty compactsubset of a metric space 〈X, d〉, then there is a function having C as its set ofminimizers that fails to be Tychonoff well-posed in the generalized sense.

The next result characterizes λ-spaces in terms of a general Cantor-typetheorem. As its validity is known in the most important special cases (see[6, 10]), it comes as no surprise.

Theorem 5.4. Let λ : 〈X, d〉 → [0,∞] be a continuous function. Then 〈X, d〉is a λ-space if and only if whenever 〈An〉 is a decreasing sequence in C0(X)

with λ(An) → 0 then⋂

n∈NAn is nonempty.

Proof. Suppose 〈X, d〉 is a λ-space and 〈An〉 is decreasing in C0(X) with λ(An) →0. For each n ∈ N, pick xn ∈ An arbitrarily. We have

0 ≤ λ(xn) ≤ supλ(a) : a ∈ An.

As λ(An) → 0, we have λ(xn) → 0, so 〈xn〉 must have a cluster point, say p.Then given ε > 0 and n0 ∈ N, ∃k ≥ n0 such that

d(xk, p) < ε ⇒ xk ∈ Sε(p) ⇒ p ∈ cl(xj : j ≥ n0) ⊆ cl

∞⋃

j=n0

Aj

⊆ An0,

because 〈An〉 is a decreasing sequence and An0is closed. Hence, p ∈ ∩n∈NAn.

Conversely, let 〈yn〉 be a sequence in 〈X, d〉 where lim λ(yn) = 0. For eachn ∈ N, put An := cl(yk : k ≥ n). Fix ε > 0; ∃n0 ∈ N such that n ≥ n0 ⇒λ(yn) < ε. As a result, ∀n ≥ n0, supλ(a) : a ∈ An ≤ ε ⇒ limλ(An) = 0.Hence

⋂∞n=1 cl(yk : k ≥ n) 6= ∅, and 〈yn〉 has a cluster point.

Lemma 5.5. Let 〈X, d〉 be a λ-space. Suppose 〈An〉 is a decreasing sequencein C0(X) with limλ(An) = 0. Then A :=

⋂

n∈NAn is nonempty and compact

and limHd(An, A) = 0.

Proof. The set A is nonempty by Theorem 5.4. Choose an arbitrary sequencex1, x2, x3, ... in A. Since λ is monotone and limλ(An) = 0, we have λ(A) = 0.Hence ∀n ∈ N, λ(xn) = 0 ⇒ 〈xn〉 has a cluster point in A because A is closed.Thus, A is compact.

Now we show lim Hd(An, A) = 0. Suppose this does not hold; then ∃ε > 0such that ∀n0 ∈ N, ∃k ≥ n0 with Hd(Ak, A) > ε. Since An0

⊇ Ak, clearlyAn0

* Sε(A). Pick ∀n ∈ N xn ∈ An \ Sε(A). Since limλ(xn) = 0, 〈xn〉 musthave a cluster point, say p. Hence

p ∈⋂

k∈N

cl(xn : n ≥ k) ⊆⋂

k∈N

Ak = A.

But ∀n ∈ N, d(xn, p) ≥ d(xn, A) ≥ ε, which is a contradiction. Thus, 〈An〉converges to A in Hausdorff distance.


Theorem 5.6. If 〈X, d〉 is complete, then the following statements are equiv-alent:

(1) 〈X, d〉 is a λ-space;(2) the measure of noncompactness functional α is continuous with respect

to λ on C0(X) : ∀ε > 0, ∃δ > 0 such that A ∈ C0(X) and λ(A) < δ ⇒α(A) < ε.

Proof. (2) ⇒ (1). Let 〈An〉 be a decreasing sequence in C0(X) with limλ(An) =0. Fix ε > 0; ∃δ > 0 such that λ(An) < δ ⇒ α(An) < ε. Since limλ(An) =0, we have lim α(An) = 0. Since X is complete, by Kuratowski’s Theorem,⋂

n∈NAn 6= ∅. Hence, by Theorem 5.4, X is a λ-space.

(1) ⇒ (2). Assume (1) holds but (2) fails, i.e., ∃ε > 0 such that given n ∈ N,∃Bn ∈ C0(X) with λ(Bn) ≤ 1

n but α(Bn) ≥ ε. Let An := x : λ(x) ≤1n and put A :=

⋂

n∈NAn which by Lemma 5.5 is nonempty and compact

and limHd(An, A) = 0. Since An ⊇ Bn, by continuity of α with respect toHausdorff distance, ∀n ∈ N, α(An) ≥ ε ⇒ α(A) ≥ ε. But α(A) = 0 as A iscompact; thus we have a contradiction.

Given an hereditary property P of open subsets of a metrizable space X ,the induced functional λP depends on the nature of the balls of the particularmetric chosen. With one choice, we might obtain a λP -space but with another,not so.

Example 5.7. Let X = 0 ∪ 1n : n ∈ N ∪ 4 − 1

n : n ∈ N as a topologicalsubspace of R, and let P (V ) be the property that V contains at most onepoint. For a particular compatible metric d, the associated functional λd

P is ofcourse the measure of isolation functional. When d is the Euclidean metric,the resulting space is not a λP -space, as λd

P (4 − 1n ) = 1

n2+n while 〈4 − 1n 〉 fails

to cluster in X . On the other hand the mapping g : X → R defined by

g(x) =

2n if x = 4 − 1n for some n

x otherwise

is a topological embedding, and this yields a metric ρ on X defined by ρ(x, w) =|g(x) − g(w)| for which 〈X, ρ〉 is a λP -space.

The next result, in the special case of UC-spaces, appears in the first JohnRainwater paper [34], a pseudonym used by mathematicians associated withthe University of Washington. In the special case of cofinally complete spaces,it is due to S. Romaguera [36].

Theorem 5.8. Let X be a metrizable topological space, and let P be an hered-itary property of open sets. The following conditions are equivalent:

(1) X has a compatible metric d such that 〈X, d〉 is a λP -space;(2) Ker(λP ) is compact.


Proof. If d is a compatible metric, let us write for the purposes of this proofSd

α(x) for the open d-ball with center x and radius α, and λdP for the induced

functional. Note that the set x ∈ X : λP (x) = 0 is well-defined, i.e., itdoes not depend on the particular metric chosen, for if ρ is another compatiblemetric, then at each x,

∀α > 0, ¬P (Sdα(x)) if and only if ∀α > 0, ¬P (Sρ

α(x)).

Let us denote this well-defined set by Ker(λP ). With this in mind, it followsfrom Theorem 5.2 that (2) is necessary for (1). For the sufficiency of (2)for (1), we use this technical fact about open covers: if X is metrizable andΩk : k ∈ N is a family of open covers of X , then there exists a compatiblemetric d for X such that ∀k ∈ N, Sd

1/k(x) : x ∈ X refines Ωk [21, p. 196].

It is possible that while compact, Ker(λP ) is empty. Then each x ∈ Xhas an open neighborhood Vx such that P (Vx). By the just-stated refinementresult, there exists a compatible metric d such that Sd

1 (x) : x ∈ X refinesVx : x ∈ X. Since P is hereditary, ∀x, λd

P (x) = supα > 0 : P (Sdα(x)) ≥ 1,

and so 〈X, d〉 is a λP -space. Otherwise, Ker(λP ) is nonempty and compact andso there is a countable family of open neighborhoods Wk : k ∈ N of Ker(λP )such that whenever V is open and Ker(λP ) ⊆ V , ∃k ∈ N with Wk ⊆ V . Again,for each x /∈ Ker(λP ), let Vx be an open neighborhood of x with P (Vx). Foreach k ∈ N, define an open cover Ωk of X as follows:

Ωk := Vx : x /∈ Wk ∪ Wk.

Choose a compatible metric d such that for each k, Sd1/k(x) : x ∈ X refines

Ωk. Now let 〈xn〉 satisfy limn→∞λdP (xn) = 0. For each k, Wk contains a tail of

〈xn〉, specifically xn ∈ Wk when λdP (xn) < 1

k . Since Wk : k ∈ N forms a base

for the neighborhoods of Ker(λP ), ∀ε > 0, ∃nε ∈ N ∀n ≥ nε, xn ∈ Sdε (Ker(λP ).

Since Ker(λP ) is compact, 〈xn〉 has a cluster point and 〈X, d〉 is a λP -space inthis second case, too.

With respect to product spaces equipped with the box metric, if we consideragain an hereditary property of open sets P , we can write a formula for λP

if the property P ”factors”, as it does in the case of the measure of isolationfunctional and the measure of local compactness functional.

Proposition 5.9. Let P1, P2 be hereditary properties of open sets in X1, X2

respectively, and P be a property of open sets in X1 × X2 such that P (U × V )if and only if both P1(U) and P2(V ). Then λP : X1 × X2 → [0,∞] can beexpressed by λP (x, y) = minλP1

(x), λP2(y).

Proof. Let x ∈ X1 and y ∈ X2. Suppose α < minλP1(x), λP2

(y). ThenP1(Sα(x)) ∧ P2(Sα(y)) ⇒ P (Sα(x, y)), and so λP (x, y) ≥ α. As a result,λP (x, y) ≥ minλP1

(x), λP2(y). Suppose β < λP (x, y). Then P (Sβ(x, y)) ⇒


P1(Sβ(x))∧P2(Sβ(y)) ⇒ λP1(x) ≥ β∧λP2

(y) ≥ β, so minλP1(x), λP2

(y) ≥ β.Hence minλP1

(x), λP2(y) ≥ λP (x, y).

Proposition 5.10. Suppose λ(x1, x2) = minλ1(x1), λ2(x2) where λ1 andλ2 are continuous, nonnegative extended real-valued functions on X1 and X2,respectively. Then λ is continuous and nonnegative, and

Ker(λ) = [Ker(λ1) × X2] ∪ [X1 × Ker(λ2)].

The next result is hinted at by a result of Hohti [23, Thm. 2.2.1] for cofinallycomplete metric spaces.

Theorem 5.11. Let 〈X1, d1〉 and 〈X2, d2〉 be metric spaces, where λ1 : X1 →[0,∞) and λ2 : X2 → [0,∞) are continuous. Consider the metric space 〈X1 ×X2, d〉, where d is the box metric, and

λ(x1, x2) = minλ1(x1), λ2(x2).

The following are equivalent:

(1) X1 × X2 is a λ-space;(2) X1 is a λ1-space, X2 is a λ2-space, and additionally both (i) Ker(λ1)

6= ∅ ⇒ X2 is compact, and (ii) Ker(λ2) 6= ∅ ⇒ X1 is compact.

Proof. (1)⇒(2): To show that X1 is a λ1-space, let 〈an〉 be a sequence in X1

where λ1(an) → 0. Consider 〈(an, c)〉 as a sequence in X1×X2, where c ∈ X2 isfixed arbitrarily. Then λ(an, c) = minλ1(an), λ2(c) → 0 because λ(an) → 0.As a result, 〈(an, c)〉 must have a cluster point (p1, c). Hence, 〈an〉 clusters. Ina similar manner, it can be shown that X2 is a λ2-space.

Suppose now Ker(λ1)6= ∅. Then ∃x ∈ X1 with λ1(x) = 0. Let 〈bn〉 be anarbitrary sequence in X2. We can then let 〈(x, bn)〉 be a sequence in X1 ×X2.Then λ(x, bn) = minλ1(x), λ2(bn) → 0 so 〈(x, bn)〉 has a cluster point (x, p3).Hence 〈bn〉 clusters ⇒ X2 compact. Similarly, it can be shown that if Ker(λ2)6=∅, then X1 is compact.

(2)⇒(1): To show X1×X2 is a λ-space, let 〈(an, bn)〉 be a sequence in X1×X2

with λ(an, bn) → 0. Consider the case where there exists a subsequence of〈an〉, say 〈an1

〉n1∈N1with N1 ⊆ N, where λ1(an1

) → 0. Then ∃N2 ⊆ N1 where〈an2

〉n2∈N2converges to a point of Ker(λ1). Since 〈bn〉 is in X2, which must

be compact, then ∃N3 ⊆ N2 such that 〈bn3〉n3∈N3

converges and 〈an3〉n3∈N3

converges. Hence, 〈(an3, bn3

)〉n3∈N3converges, which implies 〈(an, bn)〉 clusters.

In the case where there exists a subsequence of 〈bn〉, say 〈bn1〉n1∈N1

with N1 ⊆N, where λ2(bn1

) → 0, it can be similarly shown that 〈(an, bn)〉 clusters, andthis is left to the reader.

Remark 5.12. Proposition 5.10 gives an alternate justification that conditions(2i) and (2ii) are necessary in Theorem 5.11.

Example 5.13. In the case that λ1 = λ2 = the measure of local completenessfunctional, when both X1 and X2 are complete, it is clear that Ker(λ1) =


Ker(λ1) = ∅, so that X1 × X2 is complete if and only if X1 and X2 arecomplete, as we all know.

Example 5.14. In the case that λ1 = λ2 = the measure of isolation functional,condition (2) becomes X1 and X2 are both UC-spaces, and if either space haslimit points, the other must be compact.

What is most interesting about this result emerges after we take a closerlook at statement (2) of Theorem 5.11 from the perspective of mathematicallogic. Formally, statement (2) is of the form

P ∧ (Q ⇒ S) ∧ (R ⇒ T ),

which is logically equivalent to

[(P ∧ ¬Q) ∨ (P ∧ S)] ∧ [(P ∧ ¬R) ∨ (P ∧ T )].

Since conjunction is distributive over disjunction, the following four-part dis-junction is equivalent to (2):

infλ1(x) : x ∈ X1 > 0 and infλ2(x) : x ∈ X2 > 0,

or

X2 is an λ2-space, X1 is compact, and infλ1(x) : x ∈ X1 > 0,

or

X1 is an λ1-space, X2 is compact, and infλ2(x) : x ∈ X2 > 0,

or

both X1 and X2 are compact.

Thus, all factor spaces that would yield a product space that is a λ-space, whereλ is as defined in Theorem 5.11, must fall into one of these four categories.

Example 5.15. In the case that λ1 = λ2 = the measure of local compactnessfunctional, when X1 (resp. X2) is compact, then automatically λ1(x) ≡ ∞(resp. λ2(x) ≡ ∞). Thus, the final three statements of the four just listed canbe condensed down to one statement: either X1 or X2 is compact, while theother is cofinally complete. The disjunction of this statement with the first,which in this context says that both X1 and X2 are uniformly locally compact,can be seen to be equivalent to Hohti’s formulation [23].

6. λ-Subsets and Bornological Convergence

Over the last few years, there has been intense interest in bornological con-vergence of nets of sets in a metric space [12, 14, 15, 16, 30]. This was firstdescribed for nets of closed sets by Borwein and Vanderweff [17] as follows.

Definition 6.1. Let B be a bornology in metric space 〈X, d〉. We declare a net〈Ai〉i∈I of closed subsets of X B-convergent to a closed subset A of X if foreach B ∈ B and each ε > 0, we have eventually both

Ai ∩ B ⊆ Sε(A) and A ∩ B ⊆ Sε(Ai).


Notice that convergence to the empty set means that eventually the netlies outside each set in the bornology. When B = P0(X), we obtain re-stricting our attention to C0(X) convergence in Hausdorff distance becauseX ∈ P0(X). When B is the bornology of nonempty bounded subsets, weobtain Attouch-Wets convergence [2, 3, 8], also called bounded-Hausdorff con-vergence [33]. When B is the bornology of nonempty subsets with compactclosure, we obtain convergence with respect to the Fell topology [8, Theorem5.1.6], also called the topology of closed convergence [28], which for sequencesof closed sets reduces to classical Kuratowski convergence [8, Theorem 5.2.10].Recently it has been shown that convergence of linear transformations withrespect to standard topologies of uniform convergence can be understood asbornological convergence of their associated graphs [11].

Each of the bornological convergences just listed above are topological; infact, the first two are compatible with metrizable topologies on C(X). As shownin [12], those bornologies for which B-convergence is topological on C(X) arethose that are shielded from closed sets, according to the following definition.

Definition 6.2. Let B be a bornology on a metric space 〈X, d〉. We say thatB1 ∈ B is a shield for B ∈ B provided B ⊆ B1 and whenever C ∈ C0(X) isdisjoint from B1, we have Dd(B, C) > 0. We say B is shielded from closed setsprovided each B in B has a shield in the bornology.

In terms of open sets, B is shielded from closed sets if and only if givenB ∈ B ∃B1 ∈ B such that B ⊆ B1 and each neighborhood of B1 containssome ε-enlargement of B. Hence, a bornology having the property that B ∈B ⇒ ∃ε > 0 with Sε(B) ∈ B is obviously shielded from closed sets. So is abornology having a base of compact sets, as then for each B ∈ B, the compactset cl(B) serves as shield for B. More generally, whenever B is shielded fromclosed sets, then ∀B ∈ B, cl(B) ∈ B. A wealth of additional information aboutthis concept can be found in [12].

Theorem 6.3. Let λ ∈ C(X, [0,∞)) be strongly uniformly continuous on someB ∈ Bλ. Then B has a shield in Bλ.

Proof. Without loss of generality, we may assume X is not a λ-space and B isa closed λ-set. By strong uniform continuity of λ on B, ∀n ∈ N, ∃δn ∈ (0, 1

n )

such that ∀b ∈ B, ∀x ∈ X, d(x, b) < δn ⇒ |λ(b) − λ(x)| < 1n . We may also

assume that 〈δn〉 is decreasing. Let b ∈ B \ Ker(λ). There exists a smallestnb ∈ N such that 1

nb< λ(b). If x ∈ X satisfies d(x, b) < δ2nb

, then

1

2nb< λ(x) < λ(b) +

1

2nb.

Also note that λ(b) ≤ 1nb−1 , whenever nb 6= 1. Set δ(b) = δ2nb

. We claim

B1 := (Ker(λ) ∩ B) ∪⋃

b∈B\Ker(λ)

Sδ(b)(b)

is a shield for B which lies in Bλ.


We first show B1 is λ-set. Let 〈xk〉 be a sequence in B1 with λ(xk) →0. If infinitely many terms of 〈xk〉 are contained in Ker(λ) ∩ B, then 〈xk〉must cluster by the compactness of Ker(λ) ∩ B. Otherwise, by passing to asubsequence we can assume ∀k ∈ N, xk ∈

⋃

b∈B\Ker(λ) Sδ(b)(b) and λ(xk) < 12 .

Pick bk ∈ B \ Ker(λ) with xk ∈ Sδ(bk)(bk). Fix k and let’s for the moment

write n := nbk. We know that 1

2n < λ(xk), so n ≥ 2 and λ(bk) ≤ 1n−1 . Note

also 1n−1 ≤ 2

n , so

λ(bk) ≤2

n= 4 ·

1

2n< 4λ(xk).

Hence λ(bk) → 0, so 〈bk〉 has a cluster point p. Thus, p is a cluster point of〈xk〉 because δ(bk) → 0.

Now we must show whenever C ∈ C0(X) with C∩B1 = ∅, then Dd(C, B) >0. By the compactness of Ker(λ) ∩B, we find µ > 0 such that Dd(C, Ker(λ) ∩B) > 2µ. Put T1 := B∩Sµ(Ker(λ)∩B) and T2 := B \Sµ(Ker(λ)∩B), so thatT1 ∪ T2 = B. Then Dd(C, T1) ≥ µ > 0. By Theorem 4.13, there exists ε > 0such that ∀b ∈ T2, λ(b) > ε. Let k ∈ N satisfy 1

k < ε. If b ∈ T2, then λ(b) > 1k

so δ2k ≤ δ(b). Hence,

⋃

b∈T2

Sδ2k(b) ⊆

⋃

b∈T2

Sδ(b)(b) ⊆ B1.

As a result, C ∩⋃

b∈T2Sδ2k

(b) = ∅. Then Dd(C, T2) ≥ δ2k > 0. Thus,

Dd(C, B) = Dd(C, T1 ∪ T2) = minDd(C, T1), Dd(C, T2) > 0.

Corollary 6.4. Let λ : X → [0,∞) be uniformly continuous. Then Bλ isshielded from closed sets.

Example 6.5. Consider for a counterexample [0,∞) × [0,∞) equipped withthe usual metric. If λ : [0,∞) × [0,∞) → [0,∞), where λ(x, y) = xy, thenB := (x, y) : xy = 1 ∪ (x, y) : x = y and x ≤ 1 is a λ-set. SupposeB1 were a shield for B. As a result of B1 being a λ-set, ∃n ∈ N such thatB1 ∩ (x, 0) : x ≥ 0 ⊆ [0, n] × 0. Then C := [2n,∞) × 0 is closed anddisjoint from B1, but Dd(C, B) = 0. This is a contradiction. Note of coursethat λ is not strongly uniformly continuous on B.

Bornological convergence of a net 〈Ai〉i∈I of closed sets to a closed setA as determined by a bornology B can obviously be broken into two condi-tions, the first of which is called upper B-convergence, and the second lowerB-convergence [30]:

(i) ∀B ∈ B, ∀ε > 0 eventually Ai ∩ B ⊆ Sε(A), and

(ii) ∀B ∈ B, ∀ε > 0 eventually A ∩ B ⊆ Sε(Ai).

As bornologies are hereditary, evidently, (ii) is in general equivalent to

(ii′) ∀B ∈ B, B ⊆ A ⇒ ∀ε > 0, B ⊆ Sε(Ai) eventually.


As shown in [12], when the two-sided convergence is topological, condition(i) can be replaced by the following condition:

(i′) ∀B ∈ B, if Dd(B, A) > 0, then eventually Dd(B, Ai) > 0.

From condition (i′), the topology T+B

of upper B-convergence is generatedby all sets of the form A ∈ C(X) : Dd(A, B) > 0 (B ∈ B), called theupper B-proximal topology in the literature [19]. The topology T

−B

of lowerB-convergence is not so transparent. In the case that B is the bornology ofcofinally complete subsets, this was executed in [13]. Here we show that the de-scription obtained for T

−Bλ

when λ is the measure of local compactness extendsnaturally to the case when λ is a general uniformly continuous nonnegativefunctional. Our proof here is based on condition (ii′) rather than on condition(ii) as it was in the particular case addressed in [13] and seems simpler to us.

To describe a set of generators for the topology, we employ notation used in[13]: if V is a nonempty open subset of X , put V − := A ∈ C(X) : A∩V 6= ∅,and if W is a family of nonempty open subsets of X , put

W−− := A ∈ C(X) : ∃ε > 0 ∀W ∈ W, ∃aW ∈ A with Sε(aW ) ⊆ W.

Note that for a nonempty open subset V, V −− = V −.

Theorem 6.6. Let λ be a nonnegative uniformly continuous real-valued func-tion on a metric space 〈X, d〉 . Then the topology T

−Bλ

of lower Bλ-convergence

on the closed subsets of X is generated by all sets of the form V − where V is anonempty open subset of X plus all sets of the form W−− where W is a familyof nonempty open sets with inf λ(x) : x ∈ ∪W > 0.

Proof. First suppose 〈Ai〉i∈I is a net in C(X) that is lower Bλ-convergent toA. Suppose A ∈ V − where V is open. Pick a ∈ A and ε > 0 with Sε(a) ⊆ V .Since a ⊆ A, applying condition (ii′) with B = a ∈ Bλ gives eventuallyAi ∩ Sε(a) 6= ∅, so eventually Ai ∩ V 6= ∅. Next suppose A ∈ W−− whereinfλ(x) : x ∈ ∪W = µ > 0. Choose α > 0 such that ∀W ∈ W, ∃aW ∈A with Sα(aW ) ⊆ W. Since B = aW : W ∈ W ∈ Bλ and B ⊆ A, by (ii′)∃i0 ∈ I ∀i i0, B ⊆ Sα

2(Ai). Fix i i0; ∀W ∈ W, Sα

2(aW ) ∩ Ai 6= ∅, and we

conclude

Ai ∈ Sα(aW ) : W ∈ W−− ⊆ W−−.

For the converse, suppose 〈Ai〉i∈I converges to A in the topology with theprescribed set of generators. Let B1 be a fixed λ-set with B1 ⊆ A and letε > 0 be arbitrary. Put B := cl(B1) ⊆ A; it suffices to show that eventuallyB ⊆ Sε(Ai). We first consider two extreme cases for B: (1) B is compact, and(2) inf λ(b) : b ∈ B = µ > 0.


In case (1), by compactness ∃b1, b2, b3, . . . , bn in B such that B ⊆ ∪nj=1S ε

2(bj).

As b1, b2, b3, . . . , bn ⊆ A, ∀j ≤ n we have A ∈ S ε2(bj)

−, and so eventually

Ai ∈ ∩nj=1S ε

2(bj)

−. It follows that b1, b2, b3, . . . , bn ⊆ S ε2(Ai) eventually

and so B ⊆ Sε(Ai) eventually. In case (2) by uniform continuity there existsδ ∈ (0, ε) such that whenever b ∈ B and x ∈ X with d(x, b) < δ, then λ(x) > µ

2 .With W = Sδ(b) : b ∈ B, we have A ∈ W−−, so Ai ∈ W−− eventually, andwhen this occurs, B ⊆ Sδ(Ai) ⊆ Sε(Ai).

For B which does not fit into either case (1) or (2), in view of Theorem 4.13we have B ∩Ker(λ) 6= ∅, and for some ε > 0 we have B\S ε

2(B ∩Ker (λ)) 6= ∅.

By the two extreme cases just considered, eventually both

(i) B ∩ Ker(λ) ⊆ S ε2(Ai), and

(ii) B\S ε2(B ∩ Ker(λ)) ⊆ S ε

2(Ai),

and for all such i, we have B ⊆ Sε(Ai), as required.

References

[1] M.Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math.8 (1958), 11-16.

[2] H. Attouch, R. Lucchetti and R. Wets, The topology of the ρ-Hausdorff distance, Ann.Mat. Pura Appl. 160 (1991), 303–320.

[3] H. Attouch and R. Wets, Quantitative stability of variational systems: I. The epigraph-ical distance, Trans. Amer. Math. Soc. 328 (1991), 695–730.

[4] J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Marcel Dekker,New York-Basel, 1980.

[5] G. Beer, Metric spaces on which continuous functions are uniformly continuous andHausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653–658.

[6] G. Beer, More about metric spaces on which continuous functions are uniformly con-tinuous, Bull. Australian Math. Soc. 33 (1986), 397–406.

[7] G. Beer, UC spaces revisited. Amer. Math. Monthly 95 (1988), 737–739.[8] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers,

Dordrecht, Holland, 1993.[9] G. Beer, On metric boundedness structures, Set-Valued Anal. 1 (1999), 195–208.

[10] G. Beer, Between compactness and completeness, Top. Appl. 155 (2008), 503–514.[11] G. Beer, Operator topologies and graph convergence, J. Convex Anal., to appear.[12] G. Beer, C. Costantini and S. Levi, When is bornological convergence topological?,

preprint.[13] G. Beer and G. Di Maio, Cofinal completeness of the Hausdorff metric topology, preprint.[14] G. Beer and S. Levi, Pseudometrizable bornological convergence is Attouch-Wets con-

vergence, J. Convex Anal. 15 (2008), 439–453.[15] G. Beer and S. Levi, Strong uniform continuity, J. Math. Anal. Appl. 350 (2009), 568–

589.[16] G. Beer, S. Naimpally and J. Rodrıguez-Lopez, S-topologies and bounded convergences,

J. Math. Anal. Appl. 339 (2008), 542–552.[17] J. Borwein and J. Vanderwerff, Epigraphical and uniform convergence of convex func-

tions, Trans. Amer. Math. Soc. 348 (1996), 1617–1631.[18] B. Burdick, On linear cofinal completeness, Top. Proc. 25 (2000), 435–455.[19] G. Di Maio, E. Meccariello, and S. Naimpally, Uniformizing (proximal) -topologies,

Top. Appl. 137 (2004), 99–113.


[20] A. Dontchev and T. Zolezzi, Well-posed optimization problems, Lecture Notes in Math-ematics 143, Springer-Verlag, Berlin 1993.

[21] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.[22] H. Hogbe-Nlend, Bornologies and functional analysis, North-Holland, Amsterdam, 1977.[23] A. Hohti, On uniform paracompactness, Ann. Acad. Sci. Fenn. Series A Math. Diss. 36

(1981), 1–46.[24] N. Howes, Modern analysis and topology, Springer, New York, 1995.[25] S.-T. Hu, Boundedness in a topological space, J. Math Pures Appl. 228 (1949), 287–320.[26] S.-T. Hu, Intoduction to general topology, Holden-Day, San Francisco, 1966.[27] T. Jain and S. Kundu, Atsuji spaces: equivalent conditions, Topology Proc. 30 (2006),

301–325.[28] E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1984.[29] K. Kuratowski, Topology vol. 1, Academic Press, New York, 1966.[30] A. Lechicki, S. Levi and A. Spakowski, Bornological convergences, J. Math. Anal. Appl.

297 (2004), 751–770.[31] R. Lucchetti, Convexity and well-posed problems, Springer Verlag, Berlin, 2006.[32] S. Nadler and T. West, A note on Lesbesgue spaces, Topology Proc. 6 (1981), 363–369.[33] J.-P. Penot and C. Zalinescu, Bounded (Hausdorff) convergence : basic facts and ap-

plications, in Variational analysis and applications, F. Giannessi and A. Maugeri, eds.,

Kluwer Acad. Publ. Dordrecht, 2005.[34] J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math 9 (1959), 567–

570.[35] M. Rice, A note on uniform paracompactness, Proc. Amer. Math. Soc. 62 (1977), 359–

362.[36] S. Romaguera, On cofinally complete metric spaces, Q & A in Gen. Top. 16 (1998),

165–170.[37] G. Toader, On a problem of Nagata, Mathematica (Cluj) 20 (1978), 77–79.[38] W. Waterhouse, On UC spaces, Amer. Math. Monthly 72 (1965), 634–635.

Received October 2008


Gerald Beer ([email protected])Department of Mathematics, California State University Los Angeles, 5151State University Drive, Los Angeles, California 90032, USA

Manuel Segura ([email protected])Department of Mathematics, California State University Los Angeles, 5151State University Drive, Los Angeles, California 90032, USA



Volume 10, No. 1, 2009

pp. 159-171

Topologies on function spaces and hyperspaces

D. N. Georgiou∗

Abstract. Let Y and Z be two fixed topological spaces, O(Z) thefamily of all open subsets of Z, C(Y, Z) the set of all continuous mapsfrom Y to Z, and OZ(Y ) the set f−1(U) : f ∈ C(Y, Z) and U ∈ O(Z).In this paper, we give and study new topologies on the sets C(Y, Z) andOZ(Y ) calling (A,A0)-splitting and (A,A0)-admissible, where A andA0 families of spaces.

2000 AMS Classification: 54C35

Keywords: function space, hyperspace, splitting topology, admissible topol-ogy.

1. Preliminaries

Let Y and Z be two fixed topological spaces. By C(Y, Z) we denote the setof all continuous maps from Y to Z. If t is a topology on the set C(Y, Z), thenthe corresponding topological space is denoted by Ct(Y, Z).

Let X be a space. To each map g : X×Y → Z which is continuous in y ∈ Y

for each fixed x ∈ X , we associate the map g∗ : X → C(Y, Z) defined as follows:for every x ∈ X , g∗(x) is the map from Y to Z such that g∗(x)(y) = g(x, y),y ∈ Y . Obviously, for a given map h : X → C(Y, Z), the map h⋄ : X × Y → Z

defined by h⋄(x, y) = h(x)(y), (x, y) ∈ X × Y , satisfies (h⋄)∗

= h and iscontinuous in y for each fixed x ∈ X . Thus, the above association (defined in[7]) between the mappings from X × Y to Z that are continuous in y for eachfixed x ∈ X, and the mappings from X to C(Y, Z) is one-to-one.

In 1946 R. Arens [1] introduced the notion of an admissible topology: atopology t on C(Y, Z) is called admissible if the map e : Ct(Y, Z) × Y → Z,called evaluation map, defined by e(f, y) = f(y), is continuous.

In 1951 R. Arens and J. Dugundji [2] introduced the notion of a splittingtopology: a topology t on C(Y, Z) is called splitting if for every space X ,the continuity of a map g : X × Y → Z implies the continuity of the map

∗Work Supported by the Caratheodory programme of the University of Patras.

160 D. N. Georgiou

g∗ : X → Ct(Y, Z). On the set C(Y, Z) there exists the greatest splittingtopology, denoted here by tgs (see [2]). They also proved that a topology t onC(Y, Z) is admissible if and only if for every space X , the continuity of a maph : X → Ct(Y, Z) implies that of the map h⋄ : X × Y → Z

If in the above definitions it is assumed that the space X belongs to afixed class A of topological spaces, then the topology t is called A-splittingor A-admissible, respectively (see [8]). In the case where A = X we writeX-splitting (respectively, X-admissible) instead of X-splitting (respectively,X-admissible).

Let X be a space. In what follows by O(X) we denote the family of all opensubsets of X . Also, for two fixed topological spaces Y and Z we denote byOZ(Y ) the set f−1(U) : f ∈ C(Y, Z) and U ∈ O(Z).

The Scott topology Ω(Y ) on O(Y ) (see, for example, [11]) is defined as follows:a subset IH of O(Y ) belongs to Ω(Y ) if:

(α) the conditions U ∈ IH , V ∈ O(Y ), and U ⊆ V imply V ∈ IH , and(β) for every collection of open sets of Y , whose union belongs to IH , there

are finitely many elements of this collection whose union also belongsto IH .

The strong Scott topology Ωs(Y ) on O(Y ) (see [12]) is defined as follows: asubset IH of O(Y ) belongs to Ωs(Y ) if:

(α) the conditions U ∈ IH , V ∈ O(Y ), and U ⊆ V imply V ∈ IH , and(β) for every open cover of Y there are finitely many elements of this cover

whose union also belongs to IH .

The Isbell topology tIs (respectively, strong Isbell topology tsIs) on C(Y, Z)(see, for example, [13] and [12]) is the topology, which has as a subbasis thefamily of all sets of the form:

(IH, U) = f ∈ C(Y, Z) : f−1(U) ∈ IH,

where IH ∈ Ω(Y ) (respectively, IH ∈ Ωs(Y )) and U ∈ O(Z).

The compact open topology (see [7]) on C(Y, Z), denoted here by tco, is thetopology for which the family of all sets of the form

(K, U) = f ∈ C(Y, Z) : f(K) ⊆ U,

where K is a compact subset of Y and U is an open subset of Z, form a subbase.It is known that tco ⊆ tIs (see, for example, [13]).

A subset K of a space X is said to be bounded if every open cover of X hasa finite subcover for K (see [12]).

A space X is called corecompact (see [11]) if for every x ∈ X and for everyopen neighborhood U of x, there exists an open neighborhood V of x such thatthe subset V is bounded in the space U (see [11]).

Topologies on function spaces 161

Below, we give some well known results:

(1) The Isbell topology and, hence, the compact open topology, and thepoint open topology (denoted here by tpo) on C(Y, Z) are always split-ting (see, for example, [2], [3], and [13]).

(2) The compact open topology on C(Y, Z) is admissible if Y is a regularlocally compact space. In this case the compact open topology is alsothe greatest splitting topology (see [2]).

(3) The Isbell topology on C(Y, Z) is admissible if Y is a corecompactspace. In this case the Isbell topology is also the greatest splittingtopology (see, for example, [12] and [14]).

(4) A topology larger than a admissible topology is also admissible (see[2]).

(5) A topology smaller than a splitting topology is also splitting (see [2]).(6) The strong Isbell topology on C(Y, Z) is admissible if Y is a locally

bounded space (see [12]).

For a summary of all the above results and some open problems on functionspaces see [10]. Also, [4] and [5] are other papers related to this area.

In what follows if ϕ : X → Y is a map and X0 ⊆ X , then by ϕ|X0: X0 → Y

we denote the restriction of the map ϕ on the set X0. Also, if h : X × Y → Z

is a map and X0 ⊆ X , then by h|X0×Y we denote the restriction of the map h

on the set X0 × Y .

In Sections 2 and 3 we give and study new topologies on the sets C(Y, Z)and OZ(Y ) calling (A,A0)-splitting and (A,A0)-admissible, where A and A0

families of spaces.

2. (A,A0)-splitting and (A,A0)-admissible topologies on the set

C(Y, Z)

Note 1. Let A be a family of topological spaces. For every X ∈ A we denoteby X0 a subspace of X and by A0 the family of all such subspaces X0. In allpaper by (A,A0) we denote the family of all pairs (X, X0) such that X ∈ A,X0 ∈ A0, and X0 is a subspace of X.

Definition 2.1. A topology t on C(Y, Z) is called (A,A0)-splitting if for everypair (X, X0) ∈ (A,A0), the continuity of a map g : X × Y → Z implies thecontinuity of the map g∗|X0

: X0 → Ct(Y, Z), where g∗ : X → Ct(Y, Z) themap which is defined in preliminaries.

A topology t on C(Y, Z) is called (A,A0)-admissible if for every pair (X, X0) ∈(A,A0), the continuity of a map h : X → Ct(Y, Z) implies that of the maph⋄|X0×Y : X0 × Y → Z, where h⋄ : X × Y → Z the map which is defined inpreliminaries.

In the case where A = X and A0 = X0, where X0 is a subspaceof X , we write (X, X0)-splitting (respectively, (X, X0)-admissible) instead of(X, X0)-splitting (respectively, (X, X0)-admissible).

162 D. N. Georgiou

Clearly, the following theorem is true.

Theorem 2.2. The following statements are true:

(1) Every splitting (respectively, admissible) topology on C(Y, Z) is (A,A0)-splitting (respectively, (A,A0)-admissible), where A and A0 are arbi-trary families of spaces such that every element X0 ∈ A0 is a subspaceof an element X ∈ A.

(2) Every A-splitting (respectively, A-admissible) topology on C(Y, Z) is(A,A0)-splitting (respectively, (A,A0)-admissible), where A and A0

are arbitrary families of spaces such that every element X0 ∈ A0 is asubspace of an element X ∈ A.

Example 2.3.

(1) The point-open, the compact open, and the Isbell topologies are (A,A0)-splitting, where A and A0 are arbitrary families of spaces such thatevery element X0 ∈ A0 is a subspace of an element X ∈ A.

(2) If Y is a regular locally compact space, then the compact-open topologyis (A,A0)-admissible, where A and A0 are arbitrary families of spacessuch that every element X0 ∈ A0 is a subspace of an element X ∈ A.

(3) If Y is a corecompact space, then the Isbell topology is (A,A0)-admissible,where A and A0 are arbitrary families of spaces such that every elementX0 ∈ A0 is a subspace of an element X ∈ A.

(4) If Y is a locally bounded space, then the strong Isbell topology is(A,A0)-admissible, where A and A0 are arbitrary families of spacessuch that every element X0 ∈ A0 is a subspace of an element X ∈ A.

(5) Let X be a space, x0 ∈ X , X0 the subspace x0 of X , and t anarbitrary topology on C(Y, Z) which it is not X-splitting. Then, thetopology t is (X, X0)-splitting. It is clear that this topology t is notsplitting.

(6) Let X be a space, x0 ∈ X , X0 the subspace x0 of X , and t anarbitrary topology on C(Y, Z) which it is not X-admissible. Then, thetopology t is (X, X0)-admissible. It is clear that this topology t is notadmissible.


(1) A topology smaller than an (A,A0)-splitting topology is also (A,A0)-splitting.

(2) A topology larger than an (A,A0)-admissible topology is also (A,A0)-admissible.

Proof. We prove only the statement (1). The proof of (2) is similar. Let t1 bean (A,A0)-splitting topology on C(Y, Z) and t2 a topology on C(Y, Z) suchthat t2 ⊆ t1. We prove that the topology t2 is a (A,A0)-splitting topology.Indeed, let (X, X0) ∈ (A,A0) and let g : X × Y → Z be a continuous map.Since the topology t1 is (A,A0)-splitting, the map g∗|X0

: X0 → Ct1(Y, Z) iscontinuous. Also, since t2 ⊆ t1, the identical map id : Ct1(Y, Z) → Ct2(Y, Z) is


continuous. So, the map g∗|X0: X0 → Ct2(Y, Z) is continuous as a composition

of continuous maps. Thus, the topology t2 is (A,A0)-splitting.

Definition 2.5. Let (A1,A10) and (A2,A2

0) two pairs of spaces, where A1 (re-spectively, A2) and A1

0 (respectively, A20) are arbitrary families of spaces such

that every element X0 ∈ A10 (respectively, every element X0 ∈ A2

0) is a sub-space of an element X ∈ A1 (respectively, of an element X ∈ A2). We saythat the pairs (A1,A1

0) and (A2,A20) are equivalent if a topology t on C(Y, Z)

is (A1,A10)-splitting if and only if t is (A2,A2

0)-splitting, and t is (A1,A10)-

admissible if and only if t is (A2,A20)-admissible. In this case we write

(A1,A10) ∼ (A2,A2

0).

Theorem 2.6. For every pair (A,A0), where A and A0 are arbitrary familiesof spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A,there exists a pair (X(A), X(A0)), where X(A) is a space and X(A0) is asubspace of X(A) such that

(A,A0) ∼ (X(A), X(A0)).

Proof. Let T csp be the set of all topologies on C(Y, Z) which are not (A,A0)-

splitting and let T cad the set of all topologies on C(Y, Z) which are not (A,A0)-

admissible. For each t ∈ T csp there exists in (A,A0) a pair (Xsp

t , Xspt,0) such that

t is not (Xspt , X

spt,0)-splitting. Similarly, for each t ∈ T c

ad there exists in (A,A0)

a pair (Xadt , Xad

t,0) such that t is not (Xadt , Xad

t,0)-admissible. Let

A′ = Xspt : t ∈ T c

sp ∪ Xadt : t ∈ T c

ad

andA′

0 = Xspt,0 : t ∈ T c

sp ∪ Xadt,0 : t ∈ T c

ad.

Of course, we can suppose that the spaces from A′ and A′0 are pair-wise disjoint.

Let X(A) and X(A0) be the free union of all the spaces from A′ and A′0,

respectively. We prove that the pair (X(A), X(A0)) is the required pair.Let t be an (A,A0)-splitting topology on C(Y, Z). We prove that this topol-

ogy is (X(A), X(A0))-splitting. Indeed, let g : X(A)×Y → Z be a continuousmap. It suffices to prove that the map

g∗|X(A0) : X(A0) → Ct(Y, Z)

is continuous. Let X ∈ A′ ⊆ A. Then, the restriction g|X×Y of the map g onX ×Y ⊆ X(A)×Y is also a continuous map and, therefore, since the topologyt is (A,A0)-splitting we have that the map (g|X×Y )∗|X0

: X0 → Ct(Y, Z)is continuous. Since X(A0) is the free union of all the spaces from A′

0 and(g|X×Y )∗|X0

= (g∗|X(A0))|X0, it follows that the map g∗|X(A0) : X(A0) →

Ct(Y, Z) is continuous. Thus, the topology t on C(Y, Z) is (X(A), X(A0))-splitting.

Now, let t be an (X(A), X(A0))-splitting topology on C(Y, Z). We provethat t is (A,A0)-splitting. We suppose that t is not (A,A0)-splitting. Then,t ∈ T c

sp and, therefore, t is not (Xspt , X

spt,0)-splitting for some pair (Xsp

t , Xspt,0) ∈

(A,A0). Thus, there exists a continuous map g : Xspt × Y → Z such that the

164 D. N. Georgiou

map g∗|Xsp

t,0: X

spt,0 → Ct(Y, Z) is not continuous. Since the space X(A) is the

free union of all the spaces from the family A′, the map g can be extended to acontinuous map g1 : X(A) × Y → Z. Since the map g∗|Xsp

t,0is not continuous,

Xspt,0 ∈ A′

0, and the space X(A0) is the free union of all spaces from A′0 we have

that the map

g∗|X(A0) : X(A0) → Ct(Y, Z)

is not continuous, which contradicts our assumption that t is a (X(A), X(A0))-splitting topology. Thus, a topology t on C(Y, Z) is (A,A0)-splitting if and onlyif it is (X(A), X(A0))-splitting.

Similarly, a topology t on C(Y, Z) is (A,A0)-admissible if and only if is(X(A), X(A0))-admissible. Hence,

(A,A0) ∼ (X(A), X(A0)).

Theorem 2.7. There exists the greatest (A,A0)-splitting topology, where Aand A0 are arbitrary families of spaces such that every element X0 ∈ A0 is asubspace of an element X ∈ A.

Proof. Let ti : i ∈ I be the family of all (A,A0)-splitting topologies onC(Y, Z). We consider the topology t = ∨ti : i ∈ I. Clearly, t is (A,A0)-splitting and ti ⊆ t, for every i ∈ I. Thus, t is the greatest (A,A0)-splittingtopology.

Note 2. In what follows we denote by t(A,A0) the greatest (A,A0)-splittingtopology on C(Y, Z),


(1) If (A,A0) = ∪(Ai,Ai0) : i ∈ I, then

t(A,A0) = ∩t(Ai,Ai0) : i ∈ I.

(2) t(A,A0) = ∩t(X, X0) : (X, X0) ∈ (A,A0).(3) If (A,A0) = ∩(Ai,Ai

0) : i ∈ I, then

∨t(Ai,Ai0) : i ∈ I ⊆ t(A,A0).

Proof. (1) Since (A,A0) = ∪(Ai,Ai0) : i ∈ I we have that every topology

which is (A,A0)-splitting is also (Ai,Ai0)-splitting, for every i ∈ I. Thus, the

topology t(A,A0) is (Ai,Ai0)-splitting and, therefore,

t(A,A0) ⊆ t(Ai,Ai0),

for every i ∈ I. So, we have

t(A,A0) ⊆ ∩t(Ai,Ai0) : i ∈ I.

Now, we prove the converse relation, that is

∩t(Ai,Ai0) : i ∈ I ⊆ t(A,A0).


For the above relation it suffices to prove that the topology ∩t(Ai,Ai0) : i ∈ I

is (A,A0)-splitting. Let (X, X0) ∈ (A,A0) and let g : X × Y → Z be acontinuous map. We prove that the map

g∗|X0: X0 → C∩t(Ai,Ai

0):i∈I(Y, Z)

is continuous. Since (X, X0) ∈ (A,A0), there exists i ∈ I such that (X, X0) ∈(Ai,Ai

0). This means that the map

g∗|X0: X0 → Ct(Ai,Ai

0)(Y, Z)

is continuous. Also, since ∩t(Ai,Ai0) : i ∈ I ⊆ t(Ai,Ai

0), the identical map

id : Ct(Ai,Ai0)(Y, Z) → C∩t(Ai,Ai

0):i∈I(Y, Z)

is continuous. So, the map

g∗|X0: X0 → C∩t(Ai,Ai

0):i∈I(Y, Z)

is continuous as a composition of continuous maps. Thus, the topology

∩t(Ai,Ai0) : i ∈ I

is (A,A0)-splitting.(2) The proof of this is a corollary of the statement (1).(3) The proof of this follows by the fact that the topology

∨t(Ai,Ai0) : i ∈ I

is (A,A0)-splitting.

Theorem 2.9. Let t be an (A,A0)-admissible topology on C(Y, Z). If

(Ct(Y, Z), Ct(Y, Z)) ∈ (A,A0),

then t is admissible and t(A,A0) ⊆ t.

Proof. Let id ≡ h : Ct(Y, Z) → Ct(Y, Z) be the identical map. Clearly, thismap is continuous. Since

(Ct(Y, Z), Ct(Y, Z)) ∈ (A,A0)

and t is (A,A0)-admissible, the map h⋄|Ct(Y,Z) ≡ h⋄ : Ct(Y, Z) × Y → Z iscontinuous. Hence, the topology t is admissible.

Now, since the map h⋄ ≡ g : Ct(Y, Z) × Y → Z is continuous,

(Ct(Y, Z), Ct(Y, Z)) ∈ (A,A0),

and the topology t(A,A0) is (A,A0)-splitting, the map

g∗|Ct(Y,Z) = id : Ct(Y, Z) → Ct(A,A0)(Y, Z)

is also continuous. Thus, t(A,A0) ⊆ t.

Corollary 2.10. Let t be an (A,A0)-splitting and (A,A0)-admissible topologyon C(Y, Z). If (Ct(Y, Z), Ct(Y, Z)) ∈ (A,A0), then t(A,A0) = t.

Proof. By Theorem 2.9, t(A,A0) ⊆ t. Also, since the topology t is (A,A0)-splitting, t ⊆ t(A,A0). Thus, t(A,A0) = t.

166 D. N. Georgiou

Theorem 2.11. Let Y be a regular locally compact space, A the family of all Ti-spaces, i = 0, 1, 2, 3, 3 1

2 , A0 an arbitrary family of spaces containing subspacesof spaces of A, Ctco

(Y, Z) ∈ A0, and Z ∈ A. Then, we have t(A,A0) = tco =tIs.

Proof. Since Y is a regular locally compact space, the compact open topologycoincides with the Isbell topology on C(Y, Z) and it is admissible. Hence, tco

is (A,A0)-admissible. Also, the topology tco is splitting and, therefore, tco is(A,A0)-splitting. Since Z ∈ A, we have that Ctco

(Y, Z) ∈ A (see preliminaries)and, therefore, (Ctco

(Y, Z), Ctco(Y, Z)) ∈ (A,A0). Thus, by Corollary 2.10 we

have that t(A,A0) = tco.

Theorem 2.12. Let Y be a regular locally compact space, A the family ofall topological spaces whose weight is not greater than a certain fixed infinitecardinal, A0 an arbitrary family of spaces containing subspaces of spaces of A,Ctco

(Y, Z) ∈ A0, and Y, Z ∈ A. Then, we have t(A,A0) = tco = tIs.

Proof. The proof of this theorem is similar to the proof of Theorem 2.11 andfollows by Corollary 2.10 and Theorem 3.4.16 of [6].

Theorem 2.13. Let Y be a regular second-countable locally compact space, Athe family of all metrizable spaces, A0 an arbitrary family of spaces contain-ing subspaces of spaces of A, Ctco

(Y, Z) ∈ A0, and Z ∈ A. Then, we havet(A,A0) = tco = tIs.

Proof. The proof of this theorem is similar to the proof of Theorem 2.11 andfollows by Corollary 2.10 and Exercices 4.2.H and 3.4.E(c) of [6].

Theorem 2.14. Let Y be a regular locally compact Lindelof space, A the familyof all completely metrizable spaces, A0 an arbitrary family of spaces contain-ing subspaces of spaces of A, Ctco

(Y, Z) ∈ A0, and Z ∈ A. Then, we havet(A,A0) = tco = tIs.

Proof. The proof of this theorem is similar to the proof of Theorem 2.11 andfollows by Corollary 2.10 and Exercice 4.3.F(a) of [6].

Theorem 2.15. Let Y be a corecompact space, A the family of all Ti-spaces,where i = 0, 1, 2, A0 an arbitrary family of spaces containing subspaces of spacesof A, CtIs

(Y, Z) ∈ A0, and Z ∈ A. Then, we have t(A,A0) = tIs.

Proof. Since Y is corecompact, the Isbell topology tIs on C(Y, Z) is admissible.Hence the topology tIs is (A,A0)-admissible. Also, the topology tIs is splittingand, therefore, tIs is (A,A0)-splitting. Since Z ∈ A, we have that CtIs

(Y, Z) ∈A (see preliminaries) and, therefore, (CtIs

(Y, Z), CtIs(Y, Z)) ∈ (A,A0). Thus,

by Corollary 2.10 we have that t(A,A0) = tIs.

Theorem 2.16. Let Y be a corecompact space, A the family of all second-countable spaces, A0 an arbitrary family of spaces containing subspaces ofspaces of A, CtIs

(Y, Z) ∈ A0, and Y, Z ∈ A. Then, we have t(A,A0) = tIs.


Proof. The proof of this theorem is similar to the proof of Theorem 2.15 andfollows by Corollary 2.10 and the fact that CtIs

(Y, Z) ∈ A (see [12]).

3. On dual topologies

Note 3. Let Y and Z be two fixed topological spaces. By OZ(Y ) we denote theset

f−1(U) : f ∈ C(Y, Z) and U ∈ O(Z).

Let IH ⊆ OZ(Y ), H ⊆ C(Y, Z), and U ∈ O(Z). We set

(IH, U) = f ∈ C(Y, Z) : f−1(U) ∈ IH

and

(H, U) = f−1(U) : f ∈ H.

Definition 3.1. (See [9]) Let τ be a topology on OZ(Y ). The topology onC(Y, Z), for which the set

(IH, U) : IH ∈ τ, U ∈ O(Z)

is a subbasis, is called dual to τ and is denoted by t(τ).Now, let t be a topology on C(Y, Z). The topology on OZ(Y ), for which the

set

(H, U) : H ∈ t, U ∈ O(Z)

is a subbasis, is called dual to t and is denoted by τ(t).

We observe that if τ is a topology on OZ(Y ) and σ a subbasis for τ , thenthe set (IH, U) : IH ∈ σ, U ∈ O(Z) is a subbasis for t(τ) (see Lemma 2.5in [9]). Also, if t is a topology on C(Y, Z) and s a subbasis for t, then the set(H, U) : H ∈ s, U ∈ O(Z) is a subbasis for τ(t) (see Lemma 2.6 in [9]).

Note 4. Let X be a space and g : X×Y → Z a continuous map. If gx : Y → Z

is the map for which gx(y) = g(x, y), for every y ∈ Y , then by g we denote themap of X × O(Z) into OZ(Y ), for which g(x, U) = g−1

x (U) for every x ∈ X

and U ∈ O(Z).Now, let h : X → C(Y, Z) be a map. By h we denote the map of X ×O(Z)

into OZ(Y ), for which h(x, U) = (h(x))−1(U) for every x ∈ X and U ∈ O(Z).

Definition 3.2. Let τ be a topology on OZ(Y ). We say that a map M :X × O(Z) → OZ(Y ) is continuous with respect to the first variable if forevery fixed element U of O(Z), the map MU : X → (OZ(Y ), τ), for whichMU (x) = M(x, U) for every x ∈ X, is continuous.

Definition 3.3. A topology τ on OZ(Y ) is called (A,A0)-splitting if for every(X, X0) ∈ (A,A0) the continuity of a map g : X × Y → Z implies the conti-nuity with respect to the first variable of the map g|X0×O(Z) : X0 × O(Z) →(OZ(Y ), τ).

A topology τ on OZ(Y ) is called (A,A0)-admissible if for every (X, X0) ∈(A,A0) and for every map h : X → C(Y, Z) the continuity with respect to thefirst variable of the map h : X ×O(Z) → (OZ(Y ), τ) implies the continuity of

168 D. N. Georgiou

the map h⋄|X0×Y : X0 × Y → Z defined by h⋄|X0×Y (x, y) = h(x)(y), (x, y) ∈X0 × Y .

Theorem 3.4. A topology τ on OZ(Y ) is (A,A0)-splitting if and only if thetopology t(τ) on C(Y, Z) is (A,A0)-splitting.

Proof. Suppose that the topology τ on OZ(Y ) is (A,A0)-splitting, that is forevery pair (X, X0) ∈ (A,A0) the continuity of a map g : X × Y → Z impliesthe continuity with respect to the first variable of the map

g|X0×O(Z) : X0 ×O(Z) → (OZ(Y ), τ).

We prove that the topology t(τ) on C(Y, Z) is (A,A0)-splitting. Let (X, X0) ∈(A,A0) and g : X × Y → Z be a continuous map. We need to prove thatg∗|X0

: X0 → Ct(τ)(Y, Z) is a continuous map.Let x ∈ X0 and (IH, U) be an open neighborhood of (g∗|X0

)(x) in Ct(τ)(Y, Z).We must find an open neighborhood V of x in X0 such that (g∗|X0

)(V ) ⊆(IH, U). We have that ((g∗|X0

)(x))−1(U) ∈ IH . Since (g∗|X0)(x) = gx, we have

g−1x (U) ∈ IH , that is, g(x, U) ∈ IH . Since the map

g|X0×O(Z) : X0 ×O(Z) → (OZ(Y ), τ).

is continuous with respect to the first variable, the map (g|X0×O(Z))U : X0 →(OZ(Y ), τ) is continuous. Also, (g|X0×O(Z))U (x) ∈ IH . Thus, there exists anopen neighborhood V of x in X0 such that (g|X0×O(Z))U (V ) ⊆ IH .

Let x′ ∈ V . Then, (g|X0×O(Z))U (x′) ∈ IH , that is, g−1x′ (U) ∈ IH or

(g∗|X0)(x′) ∈ (IH, U). Thus, (g∗|X0

)(V ) ⊆ (IH, U), which means that themap g∗|X0

is continuous.Conversely, suppose that t(τ) is (A,A0)-splitting. We prove that τ is (A,A0)-

splitting. Let (X, X0) be an element of (A,A0) and g : X×Y → Z a continuousmap. It is sufficient to prove that g|X0×O(Z) : X0 ×O(Z) → (OZ(Y ), τ) is con-tinuous with respect to the first variable.

Let U be a fixed element of O(Z). Consider the map (g|X0×O(Z))U : X0 →

(OZ(Y ), τ). Let x ∈ X0, IH ∈ τ , and (g|X0×O(Z))U (x) = g−1x (U) ∈ IH. We need

to find an open neighborhood V of x in X0 such that (g|X0×O(Z))U (V ) ⊆ IH.

Consider the open set (IH, U) of the space Ct(τ)(Y, Z). Since

(g|X0×O(Z))U (x) = g−1x (U) ∈ IH,

we have gx ∈ (IH, U). Since t(τ) is (A,A0)-splitting, the map g∗|X0: X0 →

Ct(τ)(Y, Z) is continuous. Hence, there exists an open neighborhood V of x inX0 such that (g∗|X0

)(V ) ⊆ (IH, U).Let x′ ∈ V . Then, (g∗|X0

)(x′) = gx′ ∈ (IH, U), that is, g−1x′ (U) ∈ IH or

(g|X0×O(Z))U (x′) ∈ IH . Thus, (g|X0×O(Z))U (V ) ⊆ IH , which means that themap (g|X0×O(Z))U is continuous.

Theorem 3.5. A topology t on C(Y, Z) is (A,A0)-splitting if and only if thetopology τ(t) on OZ(Y ) is (A,A0)-splitting.

Proof. The proof of this theorem is similar to the proof of Theorem 3.4.


Example 3.6.

(1) The topologies τ(tco) and τ(tIs) are (A,A0)-splitting for every pair(A,A0). This follows by the fact that the topologies tco and tIs aresplitting and, therefore, (A,A0)-splitting.

(2) Let Z be the Sierpinski space, Ω(Y ) the Scott topology, and ΩZ(Y )the relative topology of Ω(Y ) on OZ(Y ). Then, the topology t(ΩZ(Y ))coincides with the Isbell topology on C(Y, Z). Hence, the topologyt(ΩZ(Y )) is splitting and, therefore, (A,A0)-splitting. Thus, the topol-ogy τ(t(ΩZ (Y ))) on OZ(Y ) is (A,A0)-splitting.

Theorem 3.7. A topology τ on OZ(Y ) is (A,A0)-admissible if and only if thetopology t(τ) on C(Y, Z) is (A,A0)-admissible.

Proof. Suppose that the topology τ on OZ(Y ) is (A,A0)-admissible, that isfor every space (X, X0) ∈ (A,A0) and for every map h : X → C(Y, Z) the

continuity with respect to the first variable of the map h : X × O(Z) →(OZ(Y ), τ) implies the continuity of the map h⋄|X0×Y : X0×Y → Z. We provethat t(τ) is (A,A0)-admissible. Let (X, X0) ∈ (A,A0) and h : X → Ct(τ)(Y, Z)be a continuous map. It is sufficient to prove that the map h⋄|X0×Y : X0×Y →Z is continuous. Clearly, it suffices to prove that the map h : X × O(Z) →(OZ(Y ), τ) is continuous with respect to the first variable.

Let x ∈ X , U ∈ O(Z) and IH ∈ τ such that hU (x) = h(x, U) = (h(x))−1(U) ∈IH. We prove that there exists an open neighborhood V of x in X such thathU (V ) ⊆ IH . Consider the open set (IH, U) of the space Ct(τ)(Y, Z). Then,h(x) ∈ (IH, U).

Since the map h : X → Ct(τ)(Y, Z) is continuous, there exists an openneighborhood V of x in X such that h(V ) ⊆ (IH, U).

Let x′ ∈ V . Then h(x′) ∈ (IH, U), that is (h(x′))−1(U) ∈ IH or hU (x′) =h(x′, U) ∈ IH . Thus, hU (V ) ⊆ IH , which means that hU is continuous.

Conversely, suppose that the topology t(τ) is (A,A0)-admissible. We provethat the topology τ is (A,A0)-admissible. Let (X, X0) be a pair of (A,A0) and

h : X → C(Y, Z) a map such that h : X × O(Z) → (OZ(Y ), τ) is continuouswith respect to the first variable. We need to prove that the map h⋄|X0×Y :X0 × Y → Z is continuous.

Since t(τ) is (A,A0)-admissible, it is sufficient to prove that the map h :X → Ct(τ)(Y, Z) is continuous.

Let x ∈ X , U ∈ O(Z), and IH ∈ τ such that h(x) ∈ (IH, U). Then,(h(x))−1(U) ∈ IH. Since the map hU : X → (OZ(Y ), τ) is continuous, there

exists an open neighborhood V of x in X such that hU (V ) ⊆ IH.

Let x′ ∈ V . Then, hU (x′) = (h(x′))−1(U) ∈ IH or h(x′) ∈ (IH, U). Thus,h(V ) ⊆ (IH, U), which means that the map h is continuous.

Theorem 3.8. A topology t on C(Y, Z) is (A,A0)-admissible if and only ifthe topology τ(t) on OZ(Y ) is (A,A0)-admissible.

Proof. The proof of this theorem is similar to the proof of Theorem 3.7.

170 D. N. Georgiou

Example 3.9.

(1) If Y is a regular locally compact space, then the topology τ(tco) is(A,A0)-admissible for every pair (A,A0).

(2) If Y is a corecompact space, then the topology τ(tIs) is (A,A0)-admissiblefor every pair (A,A0).

(3) If Y is a locally bounded space, then the topology τ(tsIs) is (A,A0)-admissible for every pair (A,A0).

(4) Let Ω(Y ) be the Scott topology on O(Y ). By ΩZ(Y ) we denote therelative topology of Ω(Y ) on ΩZ(Y ). If Y is corecompact, then thetopology ΩZ(Y ) is admissible (see Corollary 3.12 of [9]) and, therefore,it is (A,A0)-admissible. Thus, the topology t(ΩZ(Y )) on C(Y, Z) is(A,A0)-admissible.

Theorem 3.10. Let A and A0 are arbitrary families of spaces such that everyelement X0 ∈ A0 is a subspace of an element X ∈ A. Then in the set OZ(Y )there exists the greatest (A,A0)-splitting topology.

Proof. Let τi : i ∈ I be the set of all (A,A0)-splitting topologies on OZ(Y ).We consider the topology

τ = ∨τi : i ∈ I.

It is not difficult to prove that this topology is (A,A0)-splitting. By this fact wehave that this topology is the required greatest (A,A0)-splitting topology.

References

[1] R. Arens, A topology of spaces of transformations, Annals of Math. 47 (1946), 480–495.[2] R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951),

5–31.[3] J. Dugundji, Topology, Allyn and Bacon, Inc. Boston 1968.[4] G. Di Maio, L. Hola, D. Holy and R. McCoy, Topologies on the set space of continuous

functions, Topology Appl. 86 (1998), no. 2, 105–122.[5] G. Di Maio, E. Meccariello and S. Naimpally, Hyper-continuous convergence in function

spaces, Quest. Answers Gen. Topology 22 (2004), no. 2, 157–162.[6] R. Engelking, General Topology, Warszawa 1977.[7] R. H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945), 429-432.[8] D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos, Topologies on function spaces,

Studies in Topology VII, Zap. Nauchn. Sem. S.-Peterburg Otdel. Mat. Inst. Steklov(POMI) 208(1992), 82-97 (Russian). Translated in: J. Math. Sci., New York 81, (1996),no. 2, 2506–2514.

[9] D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos, On dual topologies, TopologyAppl. 140 (2004), 57–68.

[10] D. N. Georgiou, S.D. Iliadis and F. Mynard, Function space topologies, Open Problemsin Topology 2 (Elsevier), 15–23, 2007.

[11] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D.S. Scott, ACompendium of Continuous Lattices, Springer, Berlin-Heidelberg-New York 1980.

[12] P. Lambrinos and B. K. Papadopoulos, The (strong) Isbell topology and (weakly) con-tinuous lattices, Continuous Lattices and Applications, Lecture Notes in pure and Appl.Math. No. 101, Marcel Dekker, New York 1984, 191–211.

[13] R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lec-ture Notes in Mathematics, 1315, Springer Verlang.


[14] F. Schwarz and S. Weck, Scott topology, Isbell topology, and continuous convergence,Lecture Notes in Pure and Appl. Math. No.101, Marcel Dekker, New York 1984, 251-271.

Received February 2009

Accepted March 2009

D. N. Georgiou ([email protected])Department of Mathematics, University of Patras, 265 04 Patras, Greece

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Applied General Topology

Contents to Volume 10, Number 1 (2009)

Almost cl-supercontinuous functions. By J. K. Kohli and D.

Singh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Embedding into discretely absolutely star-Lindelof spaces

II. By Y.-K. Song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Best proximity pair theorems for relatively nonexpansive

mappings. By V. Sankar Raj and P. Veeramani . . . . . . . . . . . . . . 21

∗-half completeness in quasi-uniform spaces. By A. Andrikopou-

los . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Condensations of Cp(X) onto σ-compact spaces. By V. V. Tkachuk 39

Pointwise convergence and Ascoli theorems for nearness

spaces. By Z. Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

F-supercontinuous functions. By J. K. Kohli, D. Singh and J.

Aggarwal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Arnautov’s problems on semitopological isomorphisms. By

D. Dikranjan and A. Giordano Bruno . . . . . . . . . . . . . . . . . . . . . . . . 85

New coincidence and common fixed point theorems. By S.

L. Singh, A. Hematulin and R. Pant . . . . . . . . . . . . . . . . . . . . . . . . . 121

Well-posedness, bornologies, and the structure of metric

spaces. By G. Beer and M. Segura . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Topologies on function spaces and hyperspaces. By D. N.

Georgiou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

New coincidence and common fixed point theorems

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