THE COMBINATORICS OF OPEN COVERS

THE COMBINATORICS OF OPEN COVERS

BY MASAMI SAKAI AND MARION SCHEEPERS

The combinatorics of open covers is a study of Cantor’s diagonal argument invarious contexts. The field has its roots in a few basic selection principles that arosefrom the study of problems in analysis, dimension theory, topology and set theory.The reader will also find that some familiar works are appearing in new clothesin our survey. This is particularly the case in connection with such problems asdetermining the structure of compact scattered spaces and a number of classicalproblems in topology. We hope that the new perspective in which some of theseclassical enterprises are presented will lead to further progress. In this article wealso attempt to give the reader an overview of the problems and techniques that arecurrently fueling much of the rapidly increasing current activity in the combinatoricsof open covers.

As this is the first time that this fast emerging field is covered in the series RecentProgress in General Topology, we were not be able to survey the entire scope ofactivity in this short introductory survey. For example: Much of what is currentlydeveloping in the context of hyperspaces and weakenings of the Lindelof propertywill have to wait for another round of Recent Progress in General Topology. Weapologize to our colleagues whose works are not featured in this survey. Thereare several very useful surveys in existence that can be consulted for some insightinto the many beautiful developments we are not covering in our survey. We havecollected the ones known to us in the first part of the bibliography.

We attempt to give an exposition sensitive to separation properties. We do notmake any assumptions about separation properties of spaces except for the followingtwo: When discussing the space C(X) of continuous real-valued functions defined ina space X, we assume that X is at least T3 1

2. When discussing topological groups,

we assume that the group is at least a T0 space.Our paper is organized as follows: The first three sections introduce three selec-

tion hypotheses. This is their only purpose: to give a motivated introduction andto define some notation. We follow with three sections reporting “Recent progressin ...”.

Throughout our paper, where we mention consistency results, these should betaken as relative consistency results: We assume throughout the consistency of ZFC,Zermelo-Fraenkel set theory plus the axiom of choice.

1. Borel’s conjecture and S1(A,B).

Let R denote the set of real numbers. E. Borel [50] in his study of Lebesguemeasurability introduced the concept of a strong measure zero set of real numbers.A subset X of R is said to be strong measure zero if there is for each sequence(εn : n < ω) of positive real numbers a partition X =

⋃n<ω Xn such that for

each n, diam(Xn) < εn. Borel observed that any countable set of real numbers has1

2 BY MASAMI SAKAI AND MARION SCHEEPERS

strong measure zero. He conjectured that only countable sets of real numbers havestrong measure zero.

When Rothberger took up the study of Borel’s conjecture in [146], he formulateda topological analogue of Borel’s notion of strong measure zero. This motivates ourfirst selection principle. Let A and B be families of sets. The symbol S1(A,B)denotes the statement:

For each sequence (An : n < ω) of elements of A there is a sequence(xn : n < ω) such that each xn is an element of An, and xn : n <ω is an element of B.

Throughout we will use the symbol O to denote for a (specific) topological spacethe collection of all open covers of the space. In [146] Rothberger proved that formetrizable spaces the selection principle S1(O,O) implies Borel’s property of strongmeasure zero. The covering property S1(O,O) is now known as the Rothbergerproperty , and topological spaces with the Rothberger property are called Rothbergerspaces.

2. Menger’s conjecture and Sfin(A,B).

K. Menger [117] considered the following basis property, the Menger basis prop-erty for separable metric space (X, d):

For each basis B of X there is a sequence (Bn : n < ω) in B suchthat Bn : n < ω covers X and limn→∞ diamd(Bn) = 0.

Menger observed that every σ-compact metric space has this property, and con-jectured that if a metric space has this property, then it is σ-compact. When W.Hurewicz [95] took up the study of this conjecture he found an equivalent topologi-cal formulation for Menger’s property and for Menger’s conjecture. This motivatesour second selection principle:

Let A and B be families of sets. The symbol Sfin(A,B) denotes the statement:For each sequence (An : n < ω) of elements of A there is a sequence(Bn : n < ω) such that each Bn is a finite subset of An, and⋃Bn : n < ω is an element of B.

In [95] Hurewicz proved that for metrizable spaces Menger’s basis property isequivalent to the selection principle Sfin(O,O). The covering property Sfin(O,O)is now known as the Menger property , and topological spaces with the Mengerproperty are called Menger spaces.

Note that Sfin(O,O) is a topological, rather than metric, property. Every σ-compact topological space has the Menger property. Every topological space withthe Menger property is Lindelof. It is thus natural to ask whether theorems thathave σ-compactness as hypothesis can be strengthened by using the hypothesisSfin(O,O). It is also natural to ask whether unsolved problems using the Lindelofproperty as a hypothesis are more tractable when using the stronger hypothesisSfin(O,O).

It is known that finite powers, closed subspaces, countable unions and continu-ous images of σ-compact spaces are σ-compact, and it is natural to inquire aboutthe status of corresponding statements for Menger spaces. Rothberger spaces areMenger spaces, and thus are Lindelof spaces. It is also natural to also inquire aboutthe behavior of Rothberger spaces under products, finite powers, subspaces and soon.

THE COMBINATORICS OF OPEN COVERS 3

3. Dimension, Alexandroff’s problem and Sc(A,B).

Let n be a positive integer. There are several equivalent statements express-ing for a separable metric space that the space has Lebesgue covering dimensionn. These different statements characterizing dimension n have natural infinitarygeneralizations. The earliest such generalization is due to W. Hurewicz [98]: Aseparable metrizable space is countable dimensional if it can be written as a unionof countably many zero-dimensional subsets. Another such generalization due toP. Alexandroff [20] motivates our third selection principle. Let A and B be familiesof sets. The symbol Sc(A,B) denotes the statement:

For each sequence (An : n < ω) of elements of A there is a sequence(Bn : n < ω) such that each Bn is a disjoint refinement of the familyof sets An, and

⋃Bn : n < ω is an element of B.

For a topological space and for a positive integer n let On denote the collectionof open covers U for which |U| ≤ n. According to Alexandroff1 a metrizable spaceis weakly infinite dimensional if it has the property Sc(O2,O). And a metrizablespace is strongly infinite dimensional if it is not weakly infinite dimensional. Ev-ery countable dimensional separable metric space is weakly infinite dimensional.Alexandroff asked if the converse is true.

R.H. Bing [48] introduced the related property of screenability in his study ofmetrizability: A topological space is screenable if there is for each open cover Ua sequence (Vn : n < ω) such that each Vn is a disjoint refinement of U , and⋃

n<ω Vn is an open cover of the space. Addis and J. Gresham [18] in their studyof Alexandroff’s problem introduced a selective version of screenability, denotedSc(O,O). The property Sc(O,O) is now called selective screenability . It is knownthat the following implications hold

countable dimensional ⇒ Sc(O,O) ⇒ Sc(O2,O).Alexandroff’s problem is whether these three properties coincide in separable metriz-able spaces. R. Pol [135] solved Alexandroff’s problem by giving an example ofa compact separable metric space which is weakly infinite dimensional, but notcountable dimensional. It was soon observed that Pol’s example is in fact selec-tively screenable, raising the question whether among compact metrizable spacesweak infinite dimensionality coincides with selective screenability.

4. Recent progress on S1(A,B).

The selection principle S1(A,B) is central to several classical branches of inves-tigation. The earliest indication of this fact is Template Theorem A below. Thegame-theoretic statement in (b) and the Ramseyan statement in (c) of TemplateTheorem A are defined right after the statement of the theorem.Template Theorem A For appropriate families A and B the following three state-ments are equivalent:

(a) S1(A,B).(b) ONE has no winning strategy in the game Gω

1 (A,B).

1Strictly speaking, the definition we give here equivalent to but not he original one given byAlexandroff. This will be clarified when Alexandroff’s original definition is given below in Section6.


(c) For each positive integer k the partition relation A −→ (B)2k holds.For an ordinal number α the game Gα

1 (A,B) is played as follows: Players ONEand TWO play α innings. In inning γ < α ONE first selects an element Oγ ∈ A,and TWO responds with a Tγ ∈ Oγ . A play

O0, T0, · · · , Oγ , Tγ , · · · γ ∈ α

is won by player TWO if Tγ : γ ∈ α is an element of B: Otherwise, ONE wins.For positive integers n and k the ordinary partition symbol

A → (B)nk

abbreviates the statement:For each A ∈ A and for each function f : [A]n → 1, 2, · · · , k,there exist an i ∈ 1, 2, · · · , k and a B ∈ B such that B is a subsetof A, and f is constant of value i on [B]n.

If we take A and B to be the collection of infinite subsets of the positive integers,then this ordinary partition symbol gives the statement of Ramsey’s theorem [142].

As the investigation widened to include more families A and B, important ex-amples emerged for which these three statements are not equivalent. For otherexamples it has been found that these three equivalences can be extended to in-clude several other important mathematical statements. Our survey will report onexamples of each of these. We should mention that there are also several exampleswhere the status of the implications among (a),(b) and (c) is not yet known, orwhere the possibility of extending to a more comprehensive list of equivalences isnot known.

4.1. The Borel Conjecture. Initially investigations into the notion of a strongmeasure zero set of real numbers focused on the status of Borel’s Conjecture:BC (The Borel Conjecture) If a set of real numbers has strong measurezero, then it is countable.

Sierpinski proved [173] that the Continuum Hypothesis CH implies the existenceof uncountable strong measure zero sets of real numbers. Thus, Godel’s subsequentproof of the consistency of CH established the consistency of ¬BC, the negation ofthe Borel Conjecture. In 1976 Laver [113] proved the consistency of BC.

Theorem 4.1 (Godel, Laver). The Borel Conjecture is independent of ZFC.

This, of course, does not close the books on the Borel Conjecture, but only tellsus about its consistency status. The broader impact of the Borel Conjecture onMathematics is still emerging. Szpilrajn [179] proved that every strong measure zerometric space is zero-dimensional. Carlson [63] proved that if the Borel Conjectureholds then not only sets of real numbers, but indeed any strong measure zero metricspace is countable.

In the 1970’s, confirming a conjecture of Prikry, Galvin, Mycielski and Solovayproduced the following remarkable characterization of strong measure zero sets ofreal numbers:

Theorem 4.2 (Galvin, Mycielski and Solovay [83]). A subset X of the real line Rhas strong measure zero if, and only if, for each first category set M ⊂ R, the setX +M = x+m : x ∈ X and m ∈M is not the entire real line.


In subsequent unpublished work Galvin generalized the notion of strong measurezero to the context of topological groups as follows: Let (G,+) be a topologicalgroup. A subset X of G is said to be strong measure zero if there is for eachsequence (In : n < ω) of neighborhoods of the identity element of G, a sequence(xn : n < ω) of elements of G such that X ⊆

⋃xn + In : n < ω. This concept

was independently rediscovered years later by Kocinac who then coined the nowcommonly used term Rothberger bounded. To define this term explicitly: For atopological group (G,+), and for an open neighborhood I of the identity element,we define O(I) := x + I : x ∈ G. Then O(I) is an open cover of G. Forconvenience we define the following notation.

Onbd = O(I) : I an open neighborhood of the identity element of GAlso, for a subset Y of a topological space X we use the symbol OXY to denotethe collection of covers of Y by sets open in X. We shall assume throughout thattopological groups discussed are T0.

A subset X of a topological group (G,+) is (left) Rothberger bounded if the se-lection principle S1(Onbd,OGX) holds for the pair (G,X). And we say that thetopological group is (left) Rothberger bounded if S1(Onbd,O) holds. Some ini-tial explorations of Rothberger bounded groups indicated connections with Borel’sconjecture - [37], [186]

The following theorem gives an equivalent form of the Borel Conjecture, whichlends itself to generalization:

Theorem 4.3 ([85]). BC if, and only if, each Rothberger bounded subset of atopological group of countable weight is countable.

According to Guran [90] a topological group is ℵ0-bounded if each element ofOnbd has a countable subset which is a cover. If a topological group is Lindelof,then it is ℵ0-bounded, but the converse is not true. It is easy to find for infinitecardinals κ an ℵ0-bounded topological group of weight κ: By a theorem of Gurana topological group of weight κ is ℵ0-bounded if, and only if, it embeds into Rκ.All known ZFC examples of Rothberger bounded subsets of such groups have car-dinality at most κ. Here is a generalization of the Borel Conjecture [85]. Let κ bean infinite cardinal:

BCκ : Each Rothberger bounded subset of an ℵ0-bounded group of weightκ has cardinality at most κ.

For each infinite cardinal κ, BCκ is true for the compact Rothberger boundedsubsets of a topological group. BCℵ0 is equivalent to the classical Borel Conjecture.

Theorem 4.4 ([85]). For an uncountable cardinal κ, (1) implies (2). If BC holds,then statements (1)and (2) are equivalent.

(1) BCκ holds.(2) Each Rothberger bounded subgroup of the group2 (κ2,⊕) has cardinality at

most κ.

Like the classical Borel Conjecture, also the generalized version has broad con-sequences for the set theoretic universe. As illustration we mention two results

2The operation ⊕ on κ2 is coordinate-wise addition modulo 2.


from[85]: The consistency of ZFC + BCℵ1 implies the consistency of ZFC+ thereexists an inaccessible cardinal. If for each n we have 2ℵn < ℵω, then BCℵω impliesthat the Axiom of Projective Determinacy is true in L(R). In [85] it is also shown:If it is consistent with ZFC that there is an inaccessible cardinal that is a limit ofinaccessible cardinals, then it is consistent with ZFC that BCℵ0 + BCℵ1 hold. If itis consistent that there is a 2-huge cardinal, then it is consistent that BCℵω holds.

4.2. Rothberger spaces. The terminology “Rothberger bounded” refers to F.Rothberger who considered a close analogue of this boundedness property in theHilfssatz on page 51 of his paper [146]. In this same very influential paper Roth-berger introduced the selective covering property S1(O,O). A space is said tobe a Rothberger space if it has the property S1(O,O). The game Gω

1 (O,O) wasintroduced by Galvin [81]. Though originally introduced to analyze the Borel Con-jecture, Rothberger spaces emerged as important test cases for conjectures aboutcompact spaces and about Lindelof spaces. Characterizations of Rothberger spacesalso indicate that this concept has rich connections with several areas of mathe-matics.

Characterizations of Rothberger spacesThe following game-theoretic characterization by Pawlikowski [129] is one of the

most versatile ones for deducing other characterizations of the Rothberger property.

Theorem 4.5 (Pawlikowski). For a topological space X the following are equiva-lent:

(1) X is a Rothberger space (i.e., X has property S1(O,O)).(2) ONE has no winning strategy in the game Gω

1 (O,O).

The fact that the Rothberger property is a diagonalization property makes itperhaps less surprising that for a certain class of Lindelof spaces the Rothbergerproperty is in fact a Ramseyan partition principle. Call an open cover U of aninfinite space X an ω-cover if X is not a member of U , but for each finite setF ⊂ X there is a U ∈ U such that F ⊆ U . We define

Ω = U ∈ O : U is an open ω-cover of X.ω-covers are useful in uncovering combinatorial properties of certain classes of topo-logical spaces. One fundamental reason for this is that Ω satisfies the Ramseyanproperty (∀k ∈ N)(Ω → (Ω)1k). This fact has been exploited in various ways fortopological spaces for which each ω-cover has a countable subset which still is anω-cover. Gerlits and Nagy [87] found a characterization of topological spaces forwhich this is true for the collection of ω-covers: Define a space X to be an ε-spaceif each finite power of X is a Lindelof space.

Theorem 4.6 (Gerlits and Nagy). For a topological space X the following areequivalent:

(1) X is an ε-space.(2) Each ω-cover of X has a countable subset that is an ω-cover of X.

The following result shows that for certain spaces the Rothberger property is aRamseyan property:

Theorem 4.7 ([161]). For an ε-space X the following are equivalent:(1) X is a Rothberger space.


(2) For each positive integer k, Ω → (O)2k.

For spaces that are Rothberger in all finite powers its family of ω-covers has aparticularly rich combinatorial theory. The earliest observations in this directionare

Theorem 4.8 ([105], [148], [156], [157]). For an ε-space X the following are equiv-alent:

(1) All finite powers of X are Rothberger spaces.(2) X has the property S1(Ω,Ω).(3) ONE has no winning strategy in the game Gω

1 (Ω,Ω).(4) For all positive integers n and k, Ω → (Ω)n

k .

The equivalence of (1) and (2) in Theorem 4.8 is from [148]. The equivalenceof (2) and (3) was given in [157] while the equivalence of (2) and (4) was given in[156] and [105].

To give some of the more sophisticated Ramseyan characterizations of spaces thatare Rothberger in all finite powers we now introduce some necessary notation andterminology. For an abstract countably infinite set A define the Ellentuck topologyon [A]ℵ0 by fixing a bijective enumeration (an : n ∈ N) of A and by defining for sand T nonempty subsets of A:

s < T if: an ∈ s and am ∈ T ⇒ n < m.

Now assume that A is a countable ω-cover of an ε-space X. With the relation s < Tdefined, we define for s and T with s ⊂ A finite, T ⊂ A an ω-cover and s < T ,

[s, T ] = B ∈ [A]ℵ0 : s ⊆ B ⊆ s ∪ T and B ∈ Ω.Define a topology on [A]ℵ0 so that the family

[s,B] : s ⊂ A finite, s < B ∈ [A]ℵ0 and B ∈ Ωforms a basis for the topology. This topology is the Ellentuck topology on [A]ℵ0 .For B ⊂ A and for finite set s ⊂ A we write B|s for an ∈ B : s < an.

A subset N of a topological space is nowhere dense if there is for each nonemptyopen set U of the space a nonempty open set V ⊆ U such that V ∩ N = ∅. Asubset of a topological space is said to be meager if it is a union of countably manynowhere dense subsets. A subset of a topological space has the property of Baire ifit is the symmetric difference of some open set and some meager set.

Now define the statementE(A,B): For each countably infinite A ∈ A and for each set R ⊆[A]ℵ0

⋂B the implication (1) ⇒ (2) holds, where:

(1) R has the Baire property in the Ellentuck topology on [A]ℵ0⋂B.

(2) For each S ⊆ A with S ∈ A and each finite subset s of A,there is an infinite B ⊆ S|s with B ∈ B such that either[s,B]

⋂B ⊆ R, or else [s,B]

⋂B

⋂R = ∅.

Thus, E([N]ℵ0 , [N]ℵ0) is Ellentuck’s Theorem [74].

Theorem 4.9 ([166]). For a topological space X the following are equivalent:(1) X has the property S1(Ω,Ω).(2) E(Ω,Ω) holds for X.

The cardinality of Rothberger spaces


Arhangel’skii proved [23] that each first countable Lindelof space has cardinal-ity at most 2ℵ0 . The question of what cardinality constraints exist on Lindelofspaces for which each singleton set is a Gδ set has become known as Arhangel’skii’sproblem. For a T1 topological space X we define ψ(X), the pseudo-character ofX, to be the least infinite cardinal number κ such that each one-element subset isan intersection of at most κ open sets. The generalized version of Arhangel’skii’sproblem asks for infinite cardinal number κ what cardinality constraints there areon Lindelof spaces of pseudo-character κ. The most generally known constraint,Arhangel’skii’s upper bound theorem on the cardinality of Lindelof spaces, couldbe stated in general form as follows:

Theorem 4.10. Let κ be an infinite cardinal number. If X is a T1 Lindelof space,then the cardinality of X is less than the least measurable cardinal exceeding ψ(X).

Proof: To see this, suppose that on the contrary X has cardinality at least aslarge as the measurable cardinal µ > ψ(X). Write κ for ψ(X). For each x ∈ Xselect a family (Uα(x) : α < κ) of open subsets of X with intersection equal to x.Let Y be a subset of X such that |Y | = µ, and since µ is measurable let U be aµ-complete free ultrafilter on Y witnessing the measurability of µ.

For each x ∈ X we have

Y \ x =⋃Y \ Uα(x) : α < ψ(X)

a member of U . Thus as U is µ-complete we find for each x ∈ X an α(x) < ψ(X)for which Y \Uα(x) is a member of U . But now Uα(x)(x) : x ∈ X is an open coverof X. Consider any subset V of this open cover of X. If |V| is less than µ, then theset

Z =⋃Y \ V : V ∈ V

is, by the µ-completeness of the ultrafilter U , an element of U . Thus Z is a non-empty subset of X \ (

⋃V), implying that V is not a cover of X. 2

For a space X, let L(X) denote the least cardinal number κ such that each opencover of X has a subset of cardinality at most κ that covers X. The proof ofTheorem 4.10 shows for any T1 space X that if a measurable cardinal is larger thanboth L(X) and ψ(X), then it is larger than |X|.

The most intensively investigated case of the generalized problem is the casewhen κ = ℵ0 [22], [88], [91], [171]. It is natural to consider Arhangel’skii’s problemfor special classes of Lindelof spaces. Tall [180] launched such an investigation, andidentified the class of indestructibly Lindelof space . We shall use the followingcharacterization as our definition:

Theorem 4.11 ([170]). A Lindelof space is indestructibly Lindelof if, and only if,ONE does not have a winning strategy in the game Gω1

1 (O,O).

The importance of this class in the context of Arhangel’skii’s problem is theindependence from ZFC of upper bound 2ℵ0 on the cardinality of points Gδ inde-structibly Lindelof spaces. Tall proved in [180] that if it is consistent that there isa supercompact cardinal, then it is consistent that each points Gδ indestructiblyLindelof space has cardinality at most 2ℵ0 . This was subsequently improved to

Theorem 4.12 ([167]). If it is consistent that there is a measurable cardinal, thenit is consistent that each points Gδ indestructibly Lindelof space is of cardinality atmost 2ℵ0 .


It is natural to consider Arhangel’skii’s problem for the Rothberger spaces. ByTheorems 4.5 and 4.11 Rothberger spaces are indestructibly Lindelof, and thus theconsistency results for the indestructibly Lindelof spaces impose constraints on thecardinality of points Gδ Rothberger spaces.

Several cardinality results regarding Lindelof spaces are also consistency resultsregarding Rothberger spaces, and vice versa. One of the fundamental observationsunderlying this phenomenon is the fact that Cohen real forcing preserves variousrelevant properties while converting ground model Lindelof spaces to Rothbergerspaces. More explicitly, let κ be an infinite cardinal number and let Fn(κ, ω) be theset

p ⊂ κ× ω : p is a finite function.For p and q in Fn(κ, ω) write p < q if q ⊂ p. Then C(κ) denotes the partiallyordered set (Fn(κ, ω), <), and is known as the Cohen reals partially ordered set.

Theorem 4.13 ([170]). Let κ be an uncountable cardinal number. For each Lindelofspace X, 1C(κ) ‖− “X is Rothberger”.

Another known constraint on the cardinality of points Gδ Rothberger spaces isas follows:

Theorem 4.14 ([170]). The cardinality of a points Gδ Rothberger space is less thanthe least real-valued measurable cardinal.

If it is consistent that there exists a measurable cardinal, then it is consistentthat the upper bound of Theorem 4.14 is sharp.

Another class of cardinality constraint problems is inspired by the classical BorelConjecture. As already observed by Rothberger [146], BC implies that each metriz-able Rothberger space is countable. One can show that more generally, for eachinfinite cardinal κ the generalized Borel Conjecture BCκ implies that each T3 1

2

Rothberger space of weight at most κ, has cardinality at most κ. Little is knownabout constraints imposed on the cardinality of T1 Rothberger spaces by the gen-eralized Borel Conjectures.

Problem 1. Is it consistent that whenever X is a T1 Rothberger space, then |X| ≤w(X), the weight of X?

Like the Lindelof property, the Rothberger property is not necessarily inheritedby subspaces. A classical problem of Hajnal and Juhasz asks if each uncountableLindelof T2 space must contain a Lindelof subspace of cardinality ℵ1. Baumgartnerand Tall [4] showed that there are uncountable Lindelof T1 spaces which do nothave Lindelof subspaces of cardinality ℵ1. Koszmider and Tall [112] show it isconsistent, relative to the consistency of ZFC, that there is an uncountable LindelofT3 space which does not have a Lindelof subspace of cardinality ℵ1. By Theorem4.13 it follows that it is consistent that there is an uncountable Rothberger T3 spacewhich has no Lindelof subspace of cardinality ℵ1.

4.3. The γ-spaces and Gerlits-Nagy spaces. In their investigation of the Frechet-Urysohn property of the function space Cp(X) of continuous real-valued functionsfrom the space X to R, with the topology inherited as subspace of the Tychonoffproduct RX , Gerlits and Nagy identified the notion of a γ-space. To define thisnotion we introduce yet another class of open covers: An open cover U of a space


X is said to be a γ-cover if U is an infinite set and each infinite subset of U is anopen cover of X.

Γ = U ∈ O : U is a γ-cover of X.A topological space is said to be a γ-space if Γ is nonempty and the selectionhypothesis S1(Ω,Γ) is true for the space. Metrizable γ-spaces have been extensivelystudied since the introduction of γ-spaces in [87]. Non-metrizable γ-spaces havereceived less attention.

In the same study [87] Gerlits and Nagy introduced another class of topologicalspaces which are at least as important as the γ-spaces, but have not been studiedas intensively as the γ-spaces. To define these spaces we introduce yet another classof open covers for spaces: An open cover U of a space X is said to be groupable ifthere is a partition U =

⋃n<ω Un where each Un is finite, for m < n Um ∩ Un = ∅,

and for each x ∈ X, for all but finitely many n we have x ∈⋃Un. Observe that a

groupable cover has the property that each element of X is a member of infinitelymany elements of the cover. An open cover U is said to be large if for each x ∈ Xwe have U ∈ U : x ∈ U is infinite.

Λ = U ∈ O : U is a large cover of X.

And we also define

Ogp = U ∈ O : U is a groupable cover of X.

A topological space is said to be a Gerlits-Nagy space if it satisfies the selectionprinciple S1(Λ,Ogp). The fundamental importance of the Gerlits-Nagy spaces isstill emerging.

A somewhat recent survey of γ-spaces and Gerlits-Nagy spaces can be found in[13], but for the reader’s convenience we will repeat some items from that survey.

Many covering properties are preserved by finite unions in the sense that if Xand Y are subspaces of a space Z, and X and Y have the covering property, thenX ∪ Y has the covering property. It is known that the union of two γ-subspacesof a space need not be a γ-space [184]. F. Jordan [99, Corollary 14] proved thefollowing interesting theorem:

Theorem 4.15 (F. Jordan). Let X be a hereditarily Lindelof space in which eachopen set is the union of countably many clopen sets. If (Xn : n < ω) is a sequenceof γ-subspaces of X such that for each n < ω we have Xn ⊆ Xn+1, then

⋃n<ω Xn

is a γ-space.

Like for Rothberger spaces, there are several open problems about the cardinal-ity of γ-spaces. Criteria constraining the existence of γ-spaces of various infinitecardinalities seem more elusive. BC implies that metrizable γ-spaces are count-able. A.W. Miller [119] showed that it is consistent that each metrizable γ-spaceis countable while there are uncountable metrizable Rothberger spaces. This resultindicates that the constraints on the cardinalities of general γ-spaces might be morerestrictive than the constraints on Rothberger spaces.

Work of Gerlits and Nagy in [87] can be stated in the format

Theorem 4.16. Let κ be an infinite cardinal number. The following are equivalent:(1) κ < p.(2) Each ε-space of cardinality ≤ κ is a γ-space.


This result suggests the question whether there always is a γ-space of cardinalityp. Perhaps the most general result known for metrizable spaces is as follows:

Theorem 4.17 (Orenshtein and Tsaban [126]). If b = p, then there is a metrizableγ-space of cardinality p.

Like for Rothberger spaces, there are no upper bounds on the cardinalities ofgeneral γ-spaces. This can be seen in several ways: For each infinite cardinal numberκ the one-point compactification of a discrete space of cardinality κ is a T3 1

2γ-space

of cardinality κ. Recall that a topological space is a P-space if the intersection of anycountable family of open subsets is an open set. In [168, Proposition 6] it is shownthat there is for each infinite cardinal number κ a T3 1

2space X of cardinality κ such

that TWO has a winning strategy in the game Gω1 (O,O). But then by Theorem

1 of [87], TWO has a winning strategy in the game Gω1 (Ω,Γ), implying that X is

a γ-space. These examples are indeed Lindelof P-spaces, and thus far from beingσ-compact. Also σ-compact examples can be found:

Theorem 4.18. For each infinite cardinal κ there is a T3 12γ-space which is σ-

compact and of cardinality κ.

This can be seen by applying the preceding argument to the examples in Corol-lary 12 of [168], using Theorem 14 of [168].

It is also worth mentioning that there is a simple characterization of the propertyof being a Gerlits-Nagy space in all finite powers. For this we introduce the notionof a groupable ω-cover: An ω-cover U is said to be groupable if there is a partitionU =

⋃n<ω Un such that each Un is finite, for m < n Um ∩ Un = ∅, and for each

finite subset F of X, for all but finitely many n there is a U ∈ Un with F ⊆ U . Wedefine

Ωgp := U ∈ Ω : U is a groupable ω-cover.The following characterization of the Gerlits-Nagy property in all finite powers isanother example of Template Theorem A:

Theorem 4.19 ([111]). For an ε-space X the following are equivalent:(1) Each finite power of X has the Gerlits-Nagy property.(2) X has the property S1(Ω,Ωgp).(3) ONE has no winning strategy in the game Gω

1 (Ω,Ωgp).(4) For all positive integers n and k, Ω → (Ωgp)n

k .

Compact T2 Rothberger spaces: ConnectionsOne of the basic themes emerging from the study of Rothberger spaces is that

problems that seem hard for general Rothberger spaces are tractable for compactRothberger spaces. The structure of compact T2 Rothberger spaces is only partiallyunderstood as the following brief survey indicates. A point x in a topological spaceX is said to be an isolated point if x is an open subset of X. A topological spaceis said to be scattered if each nonempty subset of the space has an element that isan isolated point in the relative topology of that subspace.

Towards stating the next theorem, we introduce some notation that will recurin our exposition: For a T1 space X, let PR(X) be the space of all nonempty finitesubsets of X with the Pixley-Roy topology [131]: for A ∈ PR(X) and an open setU ⊂ X, define

[A,U ] = B ∈ PR(X) : A ⊂ B ⊂ U.


The family [A,U ] : A ∈ PR(X), U open in X is a base for the Pixley-Roy topol-ogy. PR(X) is called the Pixley-Roy hyperspace of a space X. It is known thatPR(X) is always zero-dimensional and completely regular: see van Douwen [71].

The following shows the central position of Rothberger spaces among compactspaces:

Theorem 4.20. Let X be a compact T2 space. The following are equivalent:(1) Cp(X) is Frechet-Urysohn. (Gerlits-Nagy [87])(2) PR(X) is normal. (Przymusinski [139])(3) Cp(X)× Y has countable tightness whenever Y does. (Uspenskij [192])(4) X is a Rothberger space.(5) TWO has a winning strategy in Gω

1 (O,O).(6) X is a γ-space.(7) X is scattered.

Note that this theorem implies that the union of countably many compact T2

γ-spaces is a γ-space. To see this, note first that the union of countably manyspaces for each of which TWO has a winning strategy in the game Gω

1 (O,O) is aspace in which TWO has a winning strategy in the game Gω

1 (O,O), and then recallthe theorem of Gerlits and Nagy that if TWO has a winning strategy in the gameGω

1 (O,O), then the space is a γ-space.Compact T2 Rothberger spaces and analysis

Studies of function spaces identified several important classes of compact spacesamong the compact T2 spaces. Compact Rothberger spaces (i.e., compact scatteredspaces) appear to have a special position among these classes. In this section webriefly survey only five classes of compact T2 spaces inspired by studies of functionspaces.

In [121], Namioka introduces the class of Radon-Nikodym compacta. There areseveral characterizations of Radon-Nikodym compactness. We use the one givenin Theorem 3.6 of [121] as definition: A compact T2 space X is Radon-Nikodymcompact if there is a set S and a subset C of the product space [0, 1]S such that

(1) C is a compact subspace of [0, 1]S and(2) For any countable subset R of S, the set CdR⊂ [0, 1]R has, in the uniform

topology 3 on [0, 1]R, a countable dense subset, and(3) X is homeomorphic to C.

Theorem 4.21 (Namioka [121] Theorem 1.4). A compact T2 Rothberger space isa Radon-Nikodym compact space.

The converse implication is false because by Theorem 5.1(b) of [121] the prod-uct of countably many Radon-Nikodym compact spaces is still a Radon-Nikodymcompact space. But a product of countable many T2 Rothberger spaces, each con-taining more than one point, contains a homeomorphic copy of the Cantor set andthus is not a Rothberger space.

A compact T2 space X is Eberlein compact if there is a set S such that X ishomeomorphic to a subspace C of [0, 1]S for which for each f ∈ C and each ε > 0the set x ∈ S : f(x) > ε is finite.

3That is, the topology induced by the metric p on [0, 1]R, where p(f, g) := sup|f(y)− g(y)| :y ∈ R.


Theorem 4.22 (Namioka [121] Theorem 1.4). Every Eberlein compact T2 space isa Radon-Nikodym compact space.

According to [175] a compact T2 space X is strong Eberlein compact if there isa set S such that X is homeomorphic to a subspace C of 0, 1S for which for eachf ∈ C the set x ∈ S : f(x) = 1 is finite4.

Theorem 4.23 (Simon, [175], Proposition 5). A T2 strong Eberlein compact spaceis a Rothberger space.

A compact T2 space X is Corson compact if there is a set S such that X ishomeomorphic to a subspace C of [0, 1]S for which for each f ∈ C and the setx ∈ S : f(x) > 0 is countable. Evidently an Eberlein compact space is Corsoncompact.

Thus the following diagram5 summarizes the implications among these concepts.CC↑

EC → RNC↑ ↑

SEC → RothbergerIt is known that a Corson compact space need not be Radon Nikodym compact:

Namioka proved

Theorem 4.24 (Namioka [121] Theorem 5.2). Each Radon-Nikodym compact spacecontains dense Gδ subspace which is metrizable in the relative topology.

Todorcevic showed that there are Corson compact spaces which do not havea dense metrizable subspace and thus by Theorem 4.24 are not Radon-Nikodymcompact, and so not Eberlein compact. The closed unit interval [0, 1] is Eberleincompact, but it is not a Rothberger space. Thus none of the implications in the leftcolumn is reversible. Since the linearly ordered compact space [0, ω1] is Rothbergercompact, but is not Corson compact, it also follows that none of the implicationsfrom left to right in this diagram is reversible.

Generalizing an earlier result of Alster [21], Orihuela, Schachermayer and Val-divia proved the interesting theorem that

Theorem 4.25 (Orihuela, Schachermayer and Valdivia [127]). A Radon-Nikodymcompact T2 space is Corson compact, if, and only if, it is Eberlein compact.

Thus, a Rothberger space is Corson compact if, and only if, it is Eberlein com-pact. Alster [21] proved that a Rothberger space is Eberlein compact if, and onlyif it is strong Eberlein compact

Compact T2 Rothberger spaces: StructureThis characterization of compact T2 Rothberger spaces makes available certain

ordinal functions that can be used to describe the structure of these spaces. For atopological space and for an ordinal α define the subspace X(α) as follows:

X(0) = X;X(α+1) = x ∈ X(α) : x not an isolated point of the space X(α);X(α) =

⋂γ<αX

(γ) when α is a limit ordinal.

4See [175], Proposition 8.5We use the following abbreviations: CC = Corson compact, EC = Eberlein compact, SEC =

strong Eberlein compact, RNC = Radon-Nikodym compact.


A space X is scattered if, and only if, there is an ordinal α > 0 for which X(α) = ∅.For a scattered space X we define the height of X as

ht(X) = minα : X(α) = ∅.When X is a scattered space of height β > 0, we define the width of X as

wd(X) = sup|X(α) \X(α+1)| : α < β.For a scattered space X of height β > 0, the cardinal sequence of X is

card(X) = 〈|X(α) \X(α+1)| : α < β〉.The structure of compact metrizable Rothberger spaces has been determined by

1920 by Mazurkiewicz and Sierpinski [116]:

Theorem 4.26 (Mazurkiewicz and Sierpinski). For compact metrizable topologicalspace X the following are equivalent:

(1) X is a Rothberger space.(2) There is a countable ordinal α such that X is homeomorphic to the ordinal

space α+ 1 endowed with the order topology.

Note that the compact metrizable Rothberger spaces have in ZFC the cardinalityrestriction imposed by BC. As noted earlier, the compact Rothberger spaces havein ZFC the cardinality restrictions imposed by the genrealized Borel Conjectures.

The structure of compact non-metrizable Rothberger spaces is less understood.

Problem 2. What are the constraints on the height of a compact T2 Rothbergerspace, given that its width is the infinite cardinal κ?

Juhasz and Weiss [102] proved

Theorem 4.27 (Juhasz and Weiss). For each ordinal α < ω2 there exists a compactT2 Rothberger space of height α and width ℵ0.

As ZFC result Theorem 4.27 is optimal: CH implies that there are no compactT2 Rothberger spaces of width ℵ0 and height ω2. This failure is not merely an issueof the cardinality of the continuum. W. Just [104] proved that it is consistent that2ℵ0 = ℵ2 and there are no compact T2 Rothberger spaces of width ℵ0 and heightω2. Baumgartner and Shelah subsequently proved that it is consistent relative tothe consistency of ZFC that there are compact T2 Rothberger spaces of width ℵ0

and height ω2. In an interesting recent publication [53] C. Brech and P. Koszmiderproduced the following strengthening of the Baumgartner Shelah result:

Theorem 4.28 (Brech and Koszmider). It is consistent, relative to the consistencyof ZFC, that there are hereditarily separable compact T2 Rothberger spaces of heightω2.

Note that the hereditary separability of Brech and Koszmider’s space impliesthat it has width ℵ0.

Martinez extended the Baumgartner-Shelah result in a different direction:

Theorem 4.29 (Martinez [115]). It is consistent, relative to the consistency ofZFC, that there is for each α < ω3 a compact T2 Rothberger space of width ℵ0 andheight α.

Miscellanea about Compact Rothberger spaces


There are many interesting facts and alternative characterizations of the compactT2 Rothberger spaces. We mention only a sample of three such items here as anillustration of the richness of the theory of compact Rothberger spaces.

Theorem 4.30 (Nyikos and Purisch [124]). A compact Rothberger space is mono-tonically normal if, and only if, it is a continuous image of a compact ordinal space.

Let (X, τ) be a topological space and let θ be a large enough regular cardinalsuch that (X, τ) ∈ Hθ and let (M,∈) be a (possibly uncountable) elementarysubmodel of Hθ such that (X, τ) ∈ M. Let XM denote the topological space withunderlying set X ∩M and with the topology generated by U ∩M : U ∈ τ ∩M.In the following theorem, which is Corollary 6.6 of [103], we assume in parts (2)and (3) that a large enough regular cardinal θ has been selected, and all mentionof elementary submodels refer to elementary submodels of Hθ:

Theorem 4.31 (Junqueira and Tall). The following are equivalent for compact T2

topological space X:(1) X is a Rothberger space.(2) There is a countable elementary submodel M such that XM is compact,

T2.(3) XM is compact T2 for every elementary submodel M such that X ∈M.

Though Eberlein compact T2 spaces need not be Rothberger spaces, they areall obtained from closed subspaces of a countable product of compact Rothbergerspaces:

Theorem 4.32 (Benyamini, Rudin and Wage [46] Lemma 1.1). For each Eberleincompact space Y there are compact Rothberger spaces Xn, n < ω, such that Y is acontinuous image of a closed subspace of Πn<ωXn.

4.4. Countable strong fan tightness. A topological space is said to be countablytight at x if any set A which has x in its closure, has a countable subset C withx in the closure of C. If a space is countably tight at each of its members, we saythat the space has countable tightness. The following notation is convenient fordiscussing tightness:

Ωx = A ⊆ X \ x : x ∈ A.The following lemma is probably well-known:

Lemma 4.33 (Folklore). Let X be a topological space such each subset is separable.Then X is countably tight at each x ∈ X.

There is a beautiful duality theory between local properties of Cp(X) and globalproperties of X. Here is an example of this phenomenon:

Theorem 4.34 ([25], [87], [140]). For a T3 12

space X, the following are equivalent:

(1) Cp(X) has countable tightness.(2) Every finite power of X is Lindelof (i.e., X is an ε-space).(3) Every open ω-cover of X contains a countable ω-cover.

The implication (1) → (2) (resp., (2) → (1)) is due to [140] (resp., [25]). Theequivalence of (2) and (3) is due to [87], recall Theorem 4.6.

If a space has the selection property S1(Ωx,Ωx) , then it is said to have countablestrong fan tightness at x . If the space has countable strong fan tightness at each


of its elements, then it is said to be countably strong fan tight . Countable strongfan tightness was introduced by M. Sakai in [148].

For certain special classes of spaces Template Theorem A does hold for the familyΩx. Function spaces provide perhaps the best known example of this fact.

Theorem 4.35. Let X be a T3 12

space. The following are equivalent:

(1) X has the property S1(Ω,Ω).(2) Cp(X) has the property S1(Ωx,Ωx) at each element x of Cp(X).(3) For each x ∈ Cp(X), ONE has no winning strategy in the game Gω

1 (Ωx,Ωx).(4) For each x ∈ Cp(X), for all positive integers n and k, Ωx → (Ωx)n

k holds.

The equivalence of the first two statements was discovered in [148]. The equiv-alence with the remaining statements were proven in [157].

Another example is given in Theorem 13B of [157]:

Theorem 4.36. For an infinite cardinal number κ the following are equivalent:(1) κ < cov(M).(2) For each T1 space X of countable tightness and for each y ∈ X such that

χ(y,X) = κ, ONE has no winning strategy in Gω1 (Ωy,Ωy).

(3) For each T1 space X of countable tightness, if y is an element of X suchthat χ(y,X) = κ, then X has countable strong fan tightness at y.

(4) For each T1 space X of countable tightness and for each y ∈ X such thatχ(y,X) = κ, for all positive integers n and k, Ωy → (Ωy)n

k holds.

But in general Template Theorem A does not hold for the family Ωx. This can beillustrated with the well-known class of spaces known as HFD spaces. A thoroughsurvey of HFD spaces can be found in [100].

Theorem 4.37 (CH). There is a T3 space which has the property S1(Ωx,Ωx) whileONE has a winning strategy in the game Gω

1 (Ωx,Ωx).

CH implies that there are HFD examples as in Theorem 4.37. On the other hand,one always has

Theorem 4.38. If X is an HFD space, then for each x ∈ X TWO has a winningstrategy in the game Gω2

1 (Ωx,Ωx).

Many examples of spaces with the property S1(Ωx,Ωx) arise through forcing:

Theorem 4.39 ([169]). Let κ be an uncountable cardinal and let X be a spacewhich is countably tight at x ∈ X. Then

1C(κ) ‖− “One has no winning strategy in the game Gω1 (Ωx,Ωx) on X.”

And according to the following interesting fact, noted in [45], Proposition 8, eachcompact countably tight space has countable strong fan tightness:

Theorem 4.40 (Bella, Matveev and Spadaro). Countably compact T2 spaces ofcountable tightness have countable strong fan tightness.

4.5. Frechet-Urysohn type properties. For a point x in a topological space Xwe define

Γx := A ⊆ X \ x : For each neighborhood U of x, A \ U is finite.Since each infinite subset of an element of Γx is an element of Γx, we may think ofΓx as the set of nontrivial sequences converging to x.


A space X is Frechet-Urysohn at x if for any set A ∈ Ωx there is a B ⊆ A withB ∈ Γx. A space X is strictly Frechet-Urysohn at x if for any sequence (An : n < ω)of elements of Ωx there are points xn ∈ An such that xn : n < ω is a member ofΓx - i.e., converges to x. A space which is (strictly) Frechet-Urysohn at each pointof the space is said to be (strictly) Frechet-Urysohn.

Observe that S1(Ωx,Γx) is strictly Frechet-Urysohn at x, which in turn impliesFrechet-Urysohn at x. In the case of Cp(X) these local properties are equivalentand precisely characterized by a selection principle of open covers of X:

Theorem 4.41 ([87]). For a T3 12

space X, let o denote the real-valued function onX which is constant of value 0. The following are equivalent:

(1) Cp(X) is Frechet-Urysohn.(2) Cp(X) is strictly Frechet-Urysohn: That is, Cp(X) has the property S1(Ωo,Γo).(3) X has the poperty S1(Ω,Γ) (i.e., X is a γ-space).(4) Every open ω-cover of X contains a γ-cover.

Among function spaces Cp(X), the k-space property, sequentiality and the Frechet-Urysohn property are equivalent [86]. Pytkeev [141] also obtained the equivalencesof those properties independently.

There is a remaining open problem from [87] related to Theorem 4.41: For asequence (An : n < ω) of subsets of a set X, we put LimAn =

⋃⋂

k≥nAk : n < ω.For a family A of subsets of a set X, L(A) denotes the smallest family of subsetsof X which contains A and is closed under the operation Lim. A space X is said tohave property (δ) (or, X is a δ-space) if for each open ω-cover U of X, X ∈ L(U)holds. Note that S1(Ω,Γ) implies property (δ) and property (δ) implies S1(O,O).A Lusin set of real numbers is a Rothberger space but does not have property (δ).BCℵ0 implies that for metrizable spaces S1(Ω,Γ) holds if, and only if, the space isa δ-space: see [87]. But the following is still open.

Problem 3 ([87]). In ZFC: Is a topological space a γ-space if, and only if it is aδ-space?

A space X is κ-Frechet-Urysohn if for any open subset U ⊂ X and any pointx ∈ U , there is a sequence xn : n < ω ⊂ U converging to x. This local propertyin Cp(X) was investigated in [150]. Continuing with alternatives of the Frechet-Urysohn property, we now introduce a property identified by Reznichenko: A spaceX has the Reznichenko property (or, is weakly Frechet-Urysohn) if: For any pointx ∈ X and any A ∈ Ωx there are pairwise disjoint finite sets Fn ⊂ A (n < ω) suchthat for every neighborhood U of x, U ∩ Fn 6= ∅ for all but finitely many n < ω.

Theorem 4.42 ([110]). For a T3 12


(1) Cp(X) has countable strong fan tightness and the Reznichenko property.(2) X has S1(Ω,Ωgp) (that is, every finite power of X is a Gerlits-Nagy space).

Nowik, Scheepers and Weiss [122] proved that a space is both a Rothbergerspace and a Hurewicz space6 if and only if it is a Gerlits-Nagy space (that is, it hasproperty (∗) introduced in Gerlits and Nagy [87]).

6We define the notion of a Hurewicz space later. These are spaces with a certain coveringproperty.


The study of local properties have also inspired some new types of coveringproperties of topological spaces. Consider for example the following characteriza-tion of the Reznichenko property: An open ω-cover U of a space X is said to beω-shrinkable if for each U ∈ U one can assign a closed subset C(U) ⊂ X such thatC(U) ⊂ U and C(U) : U ∈ U is an ω-cover of X.

Theorem 4.43 ([149]). For a T3 12


(1) Cp(X) has the Reznichenko property.(2) For each ω-shrinkable open ω-cover U of X, there are pairwise disjoint finite

subfamilies Un ⊂ U (n < ω) such that every finite subset of X is containedin some member of Un for all but finitely many n < ω.

For zero-dimensional spaces the preceding theorem has a simpler form:

Theorem 4.44 ([176]). For a zero-dimensional T3 12

space X, the following areequivalent.

(1) Cp(X) has the Reznichenko property.(2) For each clopen ω-cover U of X, there are pairwise disjoint finite subfamilies

Un ⊂ U (n < ω) such that every finite subset of X is contained in somemember of Un for all but finitely many n < ω.

A space X has the Pytkeev property if for any subset A ⊂ X and any pointx ∈ A \A, there are infinite subsets An ⊂ A (n < ω) such that every neighborhoodof x contains some An. The Pytkeev property strictly lies between subsequentialityand the Reznichenko property [114].

Theorem 4.45 ([149], [176]). For a T3 12


(1) Cp(X) has the Pytkeev property.(2) For each ω-shrinkable open ω-cover U of X, there are infinite subfamilies

Un ⊂ U (n < ω) such that ⋂Un : n < ω is an ω-cover of X.

(3) X is zero-dimensional and for each clopen ω-cover U of X, there are infinitesubfamilies Un ⊂ U (n < ω) such that

⋂Un : n < ω is an ω-cover of X.7

It is unknown whether we can remove “ω-shrinkable” in Theorem 4.43 and 4.45.We do know that if a space X has Sfin(Ω,Ω), then every open ω-cover of X isω-shrinkable [149]. The readers can refer to [150] and [185] for further results onthe Reznichenko property in Cp(X), and [189] for the Pytkeev property in Cp(X).

4.6. Quasinormal convergence and S1(Γ,Γ). We have introduced the class Γof γ-covers above. We now report on the relationship between combinatorics of γ-covers and local properties of Cp(X), and on connections with what is now knownas quasinomal convergence. The following notions were defined in [24] for i =1, 2,3 and 4: A space X has property (αi) if for each family Sn : n < ω of sequencesconverging to a point x ∈ X, there is a sequence S converging to x such that:

(α1) Sn \ S is finite for all n < ω,

(α2) Sn ∩ S is infinite for all n < ω,

(α3) Sn ∩ S is infinite for infinitely many n < ω,

7This covering property is investigated in [187].


(α4) Sn ∩ S 6= ∅ for infinitely many n < ω.For each i ∈ 1, 2, 3, (αi) implies (αi+1). Note that a space X has the property

(α2) if and only if for every family Sn : n < ω of sequences converging to a pointx ∈ X, there are points xn ∈ Sn such that xn : n < ω converges to x (this iscalled the sequence selection property in [158]). It was observed that:(i) if a space X has S1(Γ,Γ), then Cp(X) has (α2) [158], and(ii) for a space X, Cp(X) has (α2) if and only if it has (α4) [159].

Theorem 4.46 ([59], [151]). For a T3 12


(1) Cp(X) has (α2).(2) X is strongly zero-dimensional and has S1(Γclopen,Γ), where Γclopen is the

set of clopen γ-covers.

Currently the following conjecture of Scheepers is still open. No consistent coun-terexamples are known.

Conjecture 4.47 ([160]). If a perfectly normal space X is strongly zero-dimensionaland has S1(Γclopen,Γ) (i.e., Cp(X) has (α2)), then X has S1(Γ,Γ).

Perfect normality is essential in this conjecture: There is a perfect non-normalspace X1 such that Cp(X1) has (α2), but X1 does not have S1(Γ,Γ) [153]. On theother hand, under CH, there is a normal zero-dimensional non-perfect space X2

such that X2 has S1(Γclopen,Γ), but does not have S1(Γ,Γ) [49]. If the conjecturewere true for spaces of reals, then it would be true [153]. Hales gave the followinginteresting partial result regarding Conjecture 4.47

Theorem 4.48 ([92]). Let X be a perfectly normal space. If X is strongly zero-dimensional and every subspace of X has S1(Γclopen,Γ) (i.e., Cp(Y ) has (α2) forevery Y ⊂ X), then X has S1(Γ,Γ).

Theorem 4.49 ([151]). For a T3 12


(1) Cp(X) has (α1).(2) For each γ-cover Un = Un,m : m < ω (n < ω) consisting of cozero-sets

in X, there is a function ϕ ∈ ωω such that Un,m : n < ω,m ≥ ϕ(n) is aγ-cover of X.8

Hence: If a space X is perfectly normal and Cp(X) has (α1), then X has S1(Γ,Γ).Dow [72] showed that (α1) and (α2) are equivalent in Laver’s model of ZFC + BC.Therefore Conjecture 4.47 is true in Laver’s model. Some arguments on S1(Γ,Γ) andLaver’s model can be found in [120]. Bukovsky and Hales gave several equivalentconditions for Cp(X) to have (α1). One of them is: A space X has property (α0)if whenever xn,m : m < ω converges to a point x ∈ X for each n < ω, there isan unbounded non-decreasing sequence nm : m < ω such that xnm,m : m < ωconverges to x.

Theorem 4.50 ([59]). For a T3 12

space X, Cp(X) has (α1) if and only if it has(α0).

In [59] readers can find various covering properties of a perfectly normal spaceX which are equivalent to property (α1) of Cp(X).

8This covering property is a cozero set version of (α1) in Kocinac [109].


Next recall the following property due to Nyikos [123]: A space X has property(α3/2) if for each family Sn : n < ω of sequences converging to a point x ∈ Xsuch that Sn ∩ Sm = ∅ for n 6= m, there is a sequence S converging to x such thatSn \ S is finite for infinitely many n ∈ ω.

Property (α1) implies property (α3/2), and the latter property implies property(α2). In general these implications are not reversible. For the function spacesCp(X), we have:

Theorem 4.51 ([153]). For a T3 12

space X, Cp(X) has (α3/2) if and only if it has(α1).9

And now finally we turn to quasinormal convergence of functions. Let f and fn

(n < ω) be real-valued functions on a set X. According to [54] the sequence fn :n < ω converges quasinormally to f on X if there is a sequence εn : n < ω ofpositive real numbers such that lim

n→∞εn = 0 and for each x ∈ X |fn(x)−f(x)| < εn

holds for all but finitely many n < ω.This convergence was originally introduced in [65] where it was called equal

convergence. The study of thin sets in trigonometric series motivated the introduc-tion of quasinormal convergence: see [54] and [55]. Uniform convergence impliesquasinormal convergence, and quasinormal convergence implies pointwise conver-gence. None of these implications is reversible. A concrete example of pointwisenon-quasinormal convergence can be found in [55, Example 1.7].

Proposition 4.52 ([66]). Let f and fn (n < ω) be real-valued continuous functionson a space X. Then the following are equivalent:

(1) fn : n < ω converges quasinormally to f .(2) There is an increasing closed cover Ak : k < ω of X such that for each

k < ω, fn|Ak : n < ω uniformly converges to f |Ak.

The study of thin sets in trigonometric series and of Gerlits and Nagy’s γ-spacesleads the following notions: Let the symbol o denote the constant function withthe value 0. A space X is a QN-space if for each sequence fn : n < ω ⊂ Cp(X)converging to o, the sequence converges quasinormally to o. A space X is a wQN-space if for each sequence fn : n < ω ⊂ Cp(X) converging to o, the sequencecontains a subsequence which quasinormally converges to o.

Every countable space is a QN-space. Let X be a set and let κ < b be acardinal number. If X =

⋃Xα : α < κ and a sequence fn : n < ω of real-

valued functions on X converges quasinormally to f on each Xα, then the sequenceconverges quasinormally to f on X [55, Theorem 1.8]. Hence, if κ < b and a spaceis the union of κ-many QN-subspaces, then it is also a QN-space. Every γ-space is awQN-space [60], and every metrizable10 QN-space is a σ-space (i.e., every Gδ-subsetis Fσ) [143].

Surprisingly QN-spaces and wQN-spaces are related to (α1) and (α2). Scheepers[160] showed that if Cp(X) has (α2), then X is a wQN-space, and Fremlin [79]showed that the converse is also true.

Theorem 4.53 ([79], [160]). For a T3 12

space X, X is a wQN-space if and only ifCp(X) has (α2).

9This was first given for strongly zero-dimensional perfectly normal spaces X in the first versionof Tsaban and Zdomskyy [190].

10“metrizable” can be replaced by “perfectly normal” [92].


Thus Conjecture 4.47 is equivalent to the statement that “Every perfectly normalwQN-space has the property S1(Γ,Γ)”. We note that Fremlin’s s1-space in [78] alsocoincides with a wQN-space [160].

Scheepers [159] showed that if Cp(X) has (α1), then X is a QN-space, and theconverse was proved by Bukovsky, Hales [59] and Sakai [151] independently.

Theorem 4.54 ([59], [151], [159]). For a T3 12

space X, X is a QN-space if andonly if Cp(X) has (α1).

Thus wQN and QN-spaces can be characterized by the covering properties inTheorems 4.46 and 4.49 respectively. As described above, (α1) and (α2) are equiv-alent in Laver’s model. Therefore QN and wQN coincide in this model. On theother hand, let X be the space of reals constructed in [159, Theorem 6] under t = b.This space X is a wQN-space (indeed, has S1(Γ,Γ)) which is not a QN-space. Thusthe statement “QN=wQN” is independent of ZFC.

Later in this paper we introduce the Hurewicz property. For now it is sufficientto know that a subset X of the real line is Hurewicz if and only if every continuousimage of X in the Baire space NN is bounded. Using combinatorics of Borel covers,Tsaban and Zdomskyy gave the following equivalence on QN-spaces, and applied itto give an alternative proof of Theorem 4.54 (another alternative proof of Theorem4.54 is given in [62]). The implication (2)⇒(1) in Theorem 4.55 is by Corollary 5.6of [60].

Theorem 4.55 ([190]). For a perfectly normal space X, the following are equiva-lent:

(1) X is a QN-space (i.e., Cp(X) has (α1)).(2) Every Borel image of X in the Baire space NN is bounded.

Bukovsky [57] introduced and studied wQN∗ and wQN∗-spaces (also QN∗ andQN∗-spaces) to approach Conjecture 4.47: A space X is a wQN∗-space (respectivelywQN∗-space) if every sequence of real-valued lower (respectively upper) semicontin-uous functions on X converging pointwise to o contains a subsequence convergingquasinormally to o.

We define the notions of a QN∗-space and of a QN∗-space similarly. Combiningwith results in [57], [152] and [190], we finally have the following for a perfectlynormal space:

QN=QN∗=QN∗=wQN∗ → S1(Γ,Γ) =wQN∗ → wQN.

In view of this diagram Ohta and Sakai [125] introduced and studied the followingnotion. A space X has property (USC)s if for each sequence fn : n < ω of uppersemicontinuous functions from X into I = [0, 1] converging pointwise to o, thereare a subsequence nj : j < ω ⊂ ω and a sequence gj : j < ω of continuousfunctions from X into I such that fnj ≤ gj (j < ω) and gj : j < ω convergespointwise to o.

Every σ-space has (USC)s, and every wQN-space having (USC)s has S1(Γ,Γ)[125]. Hence the following problem is interesting:

Problem 4 ([125]). Does a perfectly normal wQN-space have (USC)s?

For further results on QN-spaces, wQN-spaces and their variations, readers canrefer to [56], [58], [61], [62] and references therein.


4.7. Selective versions of separability. Let X be a topological space. For ournext topic we introduce the following symbol:

D = Y ⊆ X : Y is dense in X.The strengthening S1(D,D) of separability was introduced in [162], and has beenmore intensively investigated in recent years: See for example [34], [41], [42], [43],[44], [45], [89], [144] and [154]. S1(D,D) is also referred to as R-separable by someauthors11.

The following Lemmas are probably well known:

Lemma 4.56 (Folklore). Let X be a topological space such that for each x ∈ X theproperty S1(Ωx,Ωx) holds. If X is separable, then X has the property S1(D,D).

Instances of Template Theorem A have been described for special spaces in [162].In particular:

Theorem 29 of [162] gives the following. The two cardinal functions appearing inthe statement of Theorem 4.57 are defined as follows: A collection A of nonemptyopen subsets of a topological space X is a π-base if there is for each nonemptyopen subset U of the X a member of A that is contained in U . We define π(X) =min|A| : A is a π-base of X. For a topological space X, the density of X isdenoted d(X) and is defined to be min|D| : D ⊆ X is dense in X. And thenδ(X) = minκ : d(D) ≤ κ for any dense subset D ⊆ X.

Theorem 4.57. For an infinite cardinal number κ, the following are equivalent:(1) For each T3 space X with δ(X) = ℵ0 and π(X) = κ, ONE has no winning

strategy in Gω1 (D,D) on X.

(2) κ < cov(M).(3) Each T3 space X with δ(X) = ℵ0 and π(X) = κ has the property S1(D,D).

Theorem 4.58 ([43], [44]). For a T3 12

space X, the following are equivalent:12

(1) Cp(X) satisfies S1(D,D).(2) X has a coarser second countable topology and satisfies S1(Ω,Ω).

As can be seen from Theorem 4.40 and Lemma 4.56, every separable countablytight compact space has the property S1(D,D). If a compact space has the strongerproperty of being hereditarily separable, then we have the stronger conclusion thatTWO has a winning strategy in the game Gω

1 (D,D): By the Juhasz-Shelah Theorem[101] a hereditarily separable compact space has countable π-weight. But accordingto Theorem 3 of [162]:

Theorem 4.59 ([162]). For a topological space X the following are equivalent:(1) TWO has a winning strategy in Gω

1 (D,D).(2) π(X) = ℵ0.

Thus, for example, the compact Rothberger space of Brech and Koszmider [53]described in Theorem 4.28, being compact Rothberger and hereditarily separable ineach finite power, is such that TWO has a winning strategy in the game Gω

1 (D,D)

11“R” for Rothberger. Rothberger did not consider S1(D,D). It has become customary to usethe “R” in connection with selection principles of the form S1(A,A) since Rothberger introducedthe first prototype of such a selection principle.

12This was first proved for separable metrizable spaces in [162].


on each finite power of this space. Moreover, each finite power of the space hascountable strong fan tightness.

The property S1(D,D) in the Pixley-Roy hyperspace of a space X is also in-timately related to covering properties of X: Let 2<ω be the full binary tree ofheight ω. Let X be a nonempty subset of 0, 1ω. Topologize the set 2<ω ∪X asfollows: Every point of 2<ω is isolated, and a basic neighborhood of f ∈ X is ofthe form f ∪ fdn : n ≥ k, where k < ω and fdn is the restriction of f to thedomain n. Since 2<ω ∪X is locally compact it has a T2 one-point compactification2<ω ∪X ∪ ∞. Consider the countable subspace S(X) = 2<ω ∪ ∞.

Theorem 4.60 ([154]). For a nonempty subset X ⊂ 0, 1ω, the following areequivalent:

(1) PR(S(X)) satisfies S1(D,D).(2) X satisfies S1(Ω,Ω).

Numerous examples of spaces with the property S1(D,D) also arise throughforcing:

Theorem 4.61 ([169]). If X is a separable, countably tight topological space, andif κ is an uncountable cardinal, then

1C(κ) ‖− “ ONE has no winning strategy in Gω1 (D,D) on X.”

5. Recent progress on Sfin(A,B).

The following partition relation (but not the symbol) was introduced by Baum-gartner and Taylor in their study of ultrafilters:

A → dBenk

denotes that for each element A ofA, and for each function f : [A]n → 1, 2, · · · , kthere is a set B ⊆ A, an i ∈ 1, 2, · · · , k and a partition B =

⋃n<ω Bn into finite

nonempty sets such that B ∈ B, and whenever X ⊂ B is such that |X| = n and foreach i, |X ∩Bi| ≤ 1, then f(X) = i.

The natural game associated with the selection principle Sfin(A,B) is as follows:Let an ordinal α be given: Then Gα

fin(A,B) is defined as follows: Players ONE andTWO play an inning per ordinal γ ∈ α. In inning γ ONE chooses an element Oγ

of A, and TWO responds with a finite set Tγ ⊆ Oγ . A play

O0, T0, · · · , Oγ , Tγ , γ ∈ α

is won by TWO if⋃

γ∈α Tγ is an element of B: Else, player ONE wins.As with S1(A,B), the following template theorem emerged from the earlier stud-

ies of the selection principle Sfin(A,B):

Template Theorem B The following are equivalent:

(1) Sfin(A,B).(2) ONE has no winning strategy in the game Gω

fin(A,B).(3) For each positive integer k, A → dBe2k.

Whether Template Theorem B is true depends on the families A and B, as willbe seen below.


5.1. Recent progress on Sfin(O,O). Call a topological space which has the prop-erty Sfin(O,O) a Menger space. Each σ-compact space is a Menger space, and eachMenger space is a Lindelof space. The set of irrational numbers with the topologyinherited from the real line is a Lindelof space that is not a Menger space. Asmentioned in the introductory sections of the paper, the property Sfin(O,O) madeits debut in Hurewicz’s work on a conjecture of Menger:Menger’s Conjecture: Every metric space which has the property Sfin(O,O)is σ-compact13

Sierpinski [172] pointed out that Lusin sets, which are not σ-compact, have theproperty Sfin(O,O). Since a Lusin set is not σ-compact, this indicates that CHimplies the negation of Menger’s Conjecture. Fremlin and Miller [80] proved in ZFCthat there is an uncountable set of real numbers which is not σ-compact, but is aMenger space.

Characterizations of Menger spacesHurewicz, in his analysis of metrizable Menger spaces proved two very influential

characterizations of Menger spaces.

Theorem 5.1 (Hurewicz [95]). A topological space X is a Menger space if, andonly if, ONE does not have a winning strategy in the game Gω

fin(O,O).

Theorem 5.1 has been instrumental in deriving other characterizations of Mengerspaces. Here is one of these characterizations in terms of a partition relation thatwas first considered by Baumgartner and Taylor in connection with the character-ization of P-point ultrafilters:

Theorem 5.2 ([161]). Let X be an ε-space. Then the following are equivalent:(1) X is a Menger space.(2) For each positive integer k, Ω → dOe2k holds.

Thus, Theorem 5.1 and Theorem 5.2 show that Template Theorem B is true forA = O = B.

Another type of characterization, first given by Hurewicz, has also been instru-mental in several characterizations of other properties.

Theorem 5.3 (Hurewicz, [96], Section 5). A Lindelof topological space X is aMenger space if, and only if, no continuous image of X into the product space NRis a dominating family in the eventual domination order.

Also spaces that are Menger spaces in all finite powers play a central role inseveral investigations. There is also a version of Template Theorem B for these:

Theorem 5.4 ([105], [157]). For an ε-space X the following are equivalent:(1) Each finite power of X is a Menger space.(2) X has the property Sfin(Ω,Ω).(3) ONE has no winning strategy in the game Gω

fin(Ω,Ω).(4) For all positive integers k, Ω → dΩe2k.

The equivalences appear in Theorems 3.9 and 6.2 of [105], and in Theorem 5 of[157].

13We are taking the liberty of giving an equivalent reformulation of Menger’s original conjec-ture. Menger did not mention Sfin(O,O) at all.


Relation to classical topological problemsMenger spaces have become important test cases for problems regarding the

weaker Lindelof property, or the stronger (σ-)compactness property. Though wenow survey some specific examples of this phenomenon, some examples will be leftfor later in the survey where they can be mentioned in a better context.

The D-space problemFor a topological space (X, τ) a function f : X → τ is said to be a neighborhood

assignment if for each x ∈ X we have x ∈ f(x). A topological space X is said to bea D-space if there is for each neighborhood assignment f a closed, discrete subsetD of X such that f(x) : x ∈ D is an open cover of X.

van Douwen’s Problem: Is every T3 Lindelof space a D-space?

For a long time there were no examples of classical covering properties other thanσ-compactness known to imply D. The following recent breakthrough of Aurichi [27]refocused attention on Van Douwen’s problem, and on Menger spaces:

Theorem 5.5 (Aurichi). Every T1 Menger space is a D-space.

Aurichi’s proof uses Theorem 5.1. The separation axiom T3 in van Douwen’sproblem is presumably important: Recently Soukup and Szeptycki [177] found aconsistent example of a T2 Lindelof space which is not a D-space. Using thisexample, Theorem 5.5 and Theorem 4.13 one finds that the property of not being aD-space is not preserved by generic extensions by uncountably many Cohen reals.

Productively Lindelof spacesCall a topological space X productively Lindelof if, for each Lindelof space Y theproduct X × Y is Lindelof. Every σ-compact space is productively Lindelof andCH implies that the converse is true in the class of metrizable spaces:

Theorem 5.6 (Michael, [118]). CH implies that every productively Lindelof metriz-able space is σ-compact.

In recent work Theorem 5.6 has been generalized to nonmetrizable context, asone example see [19], but even so it is not clear to what extent CH is necessary toprove that every productively Lindelof T3 space is σ-compact.

Problem 5. Is it true that every productively Lindelof T3 space is σ-compact?

With weaker hypotheses one can derive weaker, but still informative, conclusions:

Theorem 5.7 (Alas et al. [19]). b = ℵ1 implies that every T3 productively Lindelofspace is a Menger space.

The Michael space problemConclusions from the study of productively Lindelof spaces also shed light on

another classical problem. Let P denote the space of irrational numbers. A spaceis said to be a Michael space if it is Lindelof, but its product with P is not Lindelof.

The Michael Space Problem: Is there a Michael space?

There are several known circumstances that are consistent relative to the con-sistency of ZFC, and which imply the existence of a Michael space. For example,


by Theorem 5.7 b = ℵ1 implies that P is not productively Lindelof, and thus thereis a Michael space.

There are no known circumstances consistent with ZFC under which there areno Michael spaces. Below when we discuss the Hurewicz spaces, we will revisit thisproblem.

Problem 6 (Tall [182]). Let X be a productively Lindelof T3-space such that eachcompact subset of X is a Gδ subset. Is it true that if the weight of X and thecardinality of X is at most ℵ1, then X is a Menger space?

No compact space is a Michael space because compact spaces are productivelyLindelof. It is also well known that Lindelof P-spaces (which exist in abundance)are productively Lindelof. Thus, no P-space is a Michael space. F. Galvin observedthat Lindelof P-spaces are in fact γ-spaces - see [87]. One may naturally wonder if γ-spaces are productively Lindelof or at least not Michael spaces. A recent consistencyresult of Tall and Tsaban [183] Theorem 4.6 demonstrates that the answer is “no”.

Theorem 5.8 (Tall and Tsaban). Assume that b = ℵ1. Then there is a γ-space14

which is also a Michael space.

5.2. Recent progress on Sfin(Λ,Λgp). In his study of the Menger conjectureHurewicz [96] introduced another covering property that is defined as follows, andis now called the Hurewicz property : For each sequence (Un : n < ω) of open coversof the space X there is a sequence (Vn : n < ω) such that for each n, Vn is a finitesubset of Un, and for each x ∈ X, for all but finitely many n, x ∈

⋃Vn. We shall

say that a space is a Hurewicz space if it has the Hurewicz property. Hurewicz ob-served that σ-compactness implies the Hurewicz property, and that the Hurewiczproperty implies the Menger property, and conjectured:

Hurewicz Conjecture: Each metrizable Hurewicz space is σ-compact.

It was proven in [105] that the Hurewicz Conjecture is false. Hurewicz also askedif the each Menger space is a Hurewicz space, and noted in a footnote of [96] thatSieprinski communicated to him that CH implies that the answer is “no”. Onlymuch later Chaber and Pol [64] proved in ZFC that there are subspaces of thereal line that are Menger spaces but not Hurewicz spaces. Tsaban and Zdomskyysubsequently gave very useful in print treatments of ZFC examples of sets of realnumbers that are Menger spaces but not Hurewicz spaces: See for example [188]and [16].

The class of Hurewicz spaces is emerging as an important class of spaces forvarious classical topological problems. Like the Menger property, the Hurewiczproperty has become a test case for several unsolved classical problems. In addition,the Hurewicz property is important in product theorems, as will be seen below.These applications of the Hurewicz property make use of characterizations of theHurewicz property.

Characterizations of Hurewicz spacesIn [111] it was shown that

Theorem 5.9. For an ε-space X the following are equivalent:

14Note that this example is not a subspace of the real line.


(1) X is a Hurewicz space.(2) X satisfies the property Sfin(Λ,Λgp).(3) ONE has no winning strategy in the game Gω

fin(Λ,Λgp).(4) For each positive integer k, Ω → dΛgpe2k.

In particular, Template Theorem B also holds for the pair (Λ,Λgp).The following is another particularly useful characterization of Hurewicz spaces

that evolved from a similar characterization for metrizable Hurewicz spaces:

Theorem 5.10 (Tall [182]). A Lindelof T3 space X is a Hurewicz space if, and onlyif, for each Cech-complete space Y ⊇ X there is a σ-compact Z with X ⊆ Z ⊆ Y .

The following characterization of the Hurewicz property in all finite powers isanother example of Template Theorem B:

Theorem 5.11 ([111]). For an ε-space X the following are equivalent:(1) Each finite power of X is a Hurewicz space.(2) X has the property Sfin(Ω,Ωgp).(3) ONE has no winning strategy in the game Gω

fin(Ω,Ωgp).(4) For all positive integers k, Ω → dΩgpe2k.

Borel’s Conjecture and Hurewicz spacesThere are several consistent examples of Rothberger spaces that are not Hurewicz

spaces: perhaps the best known examples are Lusin sets. But all known ZFCexamples of Rothberger spaces are also Hurewicz spaces. It was not clear that thisshould be the case. The following surprising result of Tall [181] is a nice illustrationof the emerging understanding of the broader impact of Borel’s Conjecture on theset theoretic universe, and it explains why every ZFC example of a Rothberger spacemust be a Hurewicz space, and thus a Gerlits-Nagy space:

Theorem 5.12 (Tall). The Borel Conjecture implies that each Rothberger space isa Hurewicz space.

Several years before Tall’s result, Zdomskyy [193] proved the following resultwhich explained why ZFC examples of paracompact Rothberger spaces are Hurewicz.

Theorem 5.13 (Zdomskyy). u < g implies that every hereditarily Lindelof para-compact Rothberger space is Hurewicz.

Borel’s conjecture does not imply that u < g.Productively Lindelof spaces and Michael spaces again

Theorem 5.6 stated that if CH holds, then each productively Lindelof metrizablespace is a σ-compact space. Theorem 5.7 stated that under the weaker hypothesisthat b = ℵ1, every productively Lindelof T3 space is a Menger space. We now lookother hypotheses which are still weaker than CH. It is well known that add(M) ≤b ≤ d.

Theorem 5.14 (Tall [181]). d = ℵ1 implies that every productively Lindelof T3

space is a Hurewicz space.

The hypothesis of the following theorem is not exactly comparable with the pre-vious ones’ hypotheses, and indicate that there may yet be more general hypothesesfrom which it follows that productively Lindelof T3 spaces are Hurewicz spaces:


Theorem 5.15 (Tall [181]). add(M) = 2ℵ0 implies that every productively LindelofT3 space is Hurewicz.

Note that since P is not a Menger, and thus not a Hurewicz space, the hypotesesof Theorem 5.14 and Theorem 5.15 imply the existence of Michael spaces.

5.3. Countable fan tightness. A space X has countable fan tightness at x ifSfin(Ωx,Ωx) holds. This concept was introduced in [26]. If a space is countablyfan tight at each of its elements, then we say the space is countably fan tight. Notethat if a topological group (such as Cp(X)) has countable fan tightness at someelement, then it is countably fan tight. Countable strong fan tightness impliescountable fan tightness, and countable fan tightness implies countable tightness.

Theorem 5.16 ([26]). For a T3 12


(1) Cp(X) has countable fan tightness.(2) X has Sfin(Ω,Ω) (i.e., every finite power of X is Menger).

Countable fan tightness combined with the Reznichenko property in Cp(X) alsohas a satisfying characterization in terms of classical concepts:

Theorem 5.17 ([111]). For a T3 12


(1) Cp(X) has countable fan tightness and the Reznichenko property.(2) X has Sfin(Ω,Ωgp) (i.e., every finite power of X is Hurewicz).

5.4. Selective versions of separability again. The selection principle Sfin(D,D)is also called “selective separability”, and also “M-separability15” by some authors.Not every countable space is selectively separable.

Theorem 5.18 ([43], [44]). For a T3 12

space X, the following are equivalent:16

(1) Cp(X) has the property Sfin(D,D).(2) X has a coarser second countable topology and satisfies Sfin(Ω,Ω).

With S(X) and PR(S(X)) defined as before,

Theorem 5.19 ([154]). For a nonempty subset X ⊂ 0, 1ω, the following areequivalent:

(1) PR(S(X)) has the property Sfin(D,D).(2) X has the property Sfin(Ω,Ω).

Bella et al. [44] asked whether X × Y is selectively separable when X and Yare selectively separable. Several consistent counterexamples of this problem areknown, but so far no counterexample in ZFC is known. It is known that undersome set-theoretic axioms there are spaces X,Y ⊂ 0, 1ω such that X and Y haveSfin(Ω,Ω), but X × Y does not have the Menger property Sfin(O,O). Then byTheorem 5.18 (respectively 5.19), Cp(X) and Cp(Y ) (respectively PR(S(X)) andPR(S(Y ))) are selectively separable, but Cp(X)×Cp(Y ) (respectively PR(S(X))×PR(S(Y ))) is not: see [34], [144], [154]. Therefore the following problem is veryinteresting.

15“M” here refers to Menger. It has become common to call selection principles of the formSfin(A,B) an “M-property”, with reference to the Menger property Sfin(O,O) which was thefirst prototype of this type of selection principle. Note that Hurewicz actually introduced thisprototype, and poved that in metrizable spaces it is equivalent to a basis property introduced byMenger.

16This was first proved for separable metrizable spaces in [162].


Problem 7. In ZFC, are there spaces X,Y ⊂ 0, 1ω such that X and Y haveSfin(Ω,Ω), but X × Y does not have the Menger property?

6. Recent progress on Sc(A,B).

As mentioned in the introduction, for n < ω there are several characterizationsfor when a separable metric space has Lebesgue covering dimension n. Each ofthese has a natural generalization to the infinite that has been proposed as gener-alizations of the covering dimension concept to arbitrary separable metric spaces.These generalizations are topological properties and can be defined for arbitrarytopological spaces. Here we will confine our attention to the context of separablemetric spaces.

Hurewicz [97] and independently Tumarkin [191] proved that a separable met-ric space is n-dimensional if, and only if, it is a union of n + 1, but no fewer,zero-dimensional subsets. Hurewicz [98] defined: A separable metrizable space iscountable dimensional if it is a union of countably many zero-dimensional sub-sets. In 1928 Hurewicz proved that the Hilbert cube H = [0, 1]N is not countabledimensional.

For disjoint closed sets C and D in a space X, call a closed set P a partitionbetween sets C and D if there are disjoint open sets U and V with C ⊂ U , D ⊂ V ,and P = X\(U∪V ). In 1938 Eilenberg and Otto [73] proved that a separable metricspace X has dimension n if, and only if, n is minimal such that for each sequence(C1, D1), · · · , (Cn+1, Dn+1) with Ci and Di disjoint closed sets for each i, thereis a sequence (P1, · · · , Pn+1) such that Pi is a partition between Ci and Di, and⋂

i≤n+1 Pi = ∅. A finite sequence ((C1, D1), · · · , (Cm, Dm)) whose terms are pairsof nonempty, pairwise disjoint closed sets, is said to be essential if for each sequence(P1, · · · , Pn) of nonempty closed sets where for each i ≤ n Pi is a partition betweenCi and Di, we have

⋂i≤n Pi 6= ∅. Else, the sequence is said to be inessential.

Alexandroff defined a corresponding notion of infinite dimensionality (1948): X isweakly infinite dimensional17 if there is for each sequence ((Cn, Dn) : n < ∞) ofpairs of disjoint closed sets a sequence (Pn : n <∞) such that each Pn is a partitionbetween Cn and Dn, and

⋂n<∞ Pn = ∅. The notions of an essential sequence and

of an inessential sequence are defined similarly to the corresponding notions for thefinite case. When a space is not weakly infinite dimensional, it is said to be stronglyinfinite dimensional . H is strongly infinite dimensional.

The third equivalence of n-dimensionality is related to screenability: n is min-imal such that for each sequence (U1, · · · ,Un+1) of open covers, there is a se-quence (V1, · · · ,Vn+1) such that each Vi is a pairwise disjoint refinement of Ui,and

⋃i≤n+1 Vi is an open cover. This characterization appears in [18]. It is clear

that selective screenability, Sc(O,O), is a natural extension of this to the infinite.Countable dimensional implies Sc(O,O), which implies weakly infinite dimen-

sional. The second implication follows from D. Rohm’s proof that weakly infinitedimensional is equivalent to Sc(O2,O). The Alexandroff problem was whetherSc(O2,O) implies countable dimensional. As we mentioned in Sectio 3, R. Pol [135]solved this problem in 1981 by giving a compact metric space, denoted K, whichhas property Sc(O2,O) but is not countable dimensional. In fact K has propertySc(O,O).

17Note that the definition for weakly infinite dimensionality given here is the original definition,while the definition we gave earlier in Section 3 is an equivalent form of the original definition.


In [94] Haver introduced the following metric property: Metric space (X, d)has the Haver property if there is for each sequence (εn : n < ω) of positive realnumbers a corresponding sequence (Vn : n < ω) such that for each n the set Vn

is a disjoint family of open subsets of X, each such set of d-diameter less than εn,and

⋃Vn : n < ω is an open cover of X. It is easy to see that if a space has

the property Sc(O,O), then it has the Haver property in each metric generatingthe topology. E. and R. Pol have shown that conversely, if a metrizable spacehas the Haver property in all metrics generating the topology, then the space hasthe property Sc(O,O). There are examples of Haver metric spaces that are notSc(O,O).

On page 1020 of [145], Rohm asks:

Problem 8 (Rohm). (1) When is a product of two spaces an Sc(O,O) space?(2) When is a product of two spaces an Sc(O2,O) space?

Since closed subspaces inherit these properties it is necessary that each factorspace has the corresponding Sc(O,O). But it is not sufficient. The best generalpositive results known for Sc(O,O) are:

Theorem 6.1 (Hattori-Yamada -[93], Rohm-[145]). If X is σ-compact and both Xand Y have property Sc(O,O), then so does their product.

and

Theorem 6.2 (Babinkostova [31]). If X and Y are metrizable spaces, each withthe property Sc(O,O), and if X × Y has the Hurewicz property, then X × Y hasproperty Sc(O,O).

E. and R. Pol proved that in Theorem 6.2 the hypothesis that X × Y has theHurewicz property cannot be weakened to the hypothesis thatX×Y has the Mengerproperty:

Theorem 6.3 (E. and R. Pol [134]). (add(M) = 2ℵ0) There is a separable metricspace X with property Sc(O,O) such that X2 has Sfin(O,O) but not the Haverproperty.

σ-compactness is stronger than having the Hurewicz property in all finite powers.Thus, for finite powers the following is a real improvement on the Hattori-Yamada,Rohm Theorem:

Theorem 6.4 ([31]). If X has Sfin(Ω,Ωgp), then it has Sc(O,O) if, and only if,it has Sc(O,O) in all finite powers.

Theorem 6.4 suggests a potential improvement of Theorem 6.1:

Problem 9. Can σ-compactness in Theorem 6.1 be replaced by Sfin(Ω,Ωgp)?

We suspect that the answer is “No”. We have less intuition about the nextquestion:

Problem 10. Can Sfin(Ω,Ωgp) in Theorem 6.4 be replaced with Sfin(Ω,Ogp)?

The following reasoning suggests that “Yes” to Problem 10 might be possible:

Theorem 6.5 ([31]). Any X with the Haver property and Sfin(Ω,Ogp) has theHaver property in all finite powers.


By Theorem 6.3 the hypothesis Sfin(Ω,Ogp) in Theorem 6.5 cannot be weakenedto Sfin(Ω,O). Although the Haver property does not imply Sc(O,O), we have:

Theorem 6.6 ([31]). If X has the Haver property and Sfin(Ω,Ogp), then it hasproperty Sc(O,O).

E. and R. Pol showed:

Theorem 6.7 (E. Pol and R. Pol, [134]). (add(M) = 2ℵ0) There is a separa-ble metric space X with the Haver property and Sfin(O,O), but not the propertySc(O,O).

We conjecture that a positive answer to Problem 10 is independent of ZFC. E.Pol investigated replacing the hypothesis σ-compact + Sc(O,O) in Theorem 6.1with another strengthening of Sc(O,O) in one of the factors: Finite dimensionalspaces have Sc(O,O) in a strong sense. E. Pol showed that even strengthening to“zero-dimensional” is not sufficient - [132], [133]:

Theorem 6.8 (E. Pol). (1) There are separable metric spaces X and Y withX zero-dimensional and Y with property Sc(O,O), but X×Y not Sc(O2,O).

(2) (CH) X can be taken to be P, and Y can be taken to have property Sc(O,O).

Which zero-dimensional spaces would give a positive product theorem? Figure4 indicates some strengthenings of zero-dimensional. Here, SMZ denotes “strongmeasure zero”.

zero-dimensional(Negative, E. Pol)

↑S1(Ω,Ogp) −→ S1(Ω,O) −→ SMZ

↑ ↑ ↑S1(Ω,Γ) −→ S1(Ω,Ωgp) −→ S1(Ω,Ω) −→ (∀n)SMZn

↑countable(Positive)

Figure 4: Strengthenings of zero-dimensionalBC implies that any of the seven alternatives between the two boxed items is a

successful replacement for σ-compact +Sc(O,O) in Theorem 6.1. We expect thatnot all of them provably are. Using game theory we identified a collection of X’sin S1(Ω,O) which yields a positive product theorem, but we had to strengthen thehypothesis on Y : We proved in [39]:

Theorem 6.9. Let Y be a space with property Sc(O,O) and Sfin(Ω,Ogp). If TWOhas a winning strategy in Gω+ω

1 (O,O) on X, then X × Y has Sc(O,O).

We can prove this theorem when TWO has a winning strategy in Gω·n1 (O,O) on

X, for a finite n. But getting past ω2 poses new problems. In fact, the proof ofthis theorem applies to all the sets of reals Baldwin constructed in [40] using CH:Each countable limit ordinal is tpS1(O,O)(X) for some such X.

It seems plausible that in Theorem 6.9 the hypothesis that “TWO has a winningstrategy in Gω+ω

1 (O,O)” can be weakened to “X and Y have Sc(O,O) and forsome countable ordinal α TWO has a winning strategy in Gα

fin(O,O) on X”. This


is true when α = ω because then by a theorem of Telgarsky we have the hypothesesof Theorem 6.1. But even for α = ω + ω we don’t know if this generalization iscorrect. Here is a concrete test-case:

Problem 11. If X is a Sierpinski set and Y has Sc(O,O) and Sfin(Ω,Ogp), thendoes X × Y have property Sc(O,O)?

The best result known for the product theory for Sc(O2,O) is:

Theorem 6.10 (Hattori-Yamada-[93], Rohm-[145]). If X is σ-compact and hasproperty Sc(O2,O), and if Y has property Sc(O,O), then X × Y has Sc(O2,O).

The following problem, mentioned in several places in the literature -[75], p. 301,[93], p. 556, [136], p. 98, [138], p. 86, is still unsolved:

Problem 12. If X and Y are both compact and have property Sc(O2,O), then doesX × Y have property Sc(O2,O)?

The S1- and Sfin- versions of Rohm’s Problem has not been studied for theclassical selection principles! Most results reported in the literature state thatcertain classical selection principles are not preserved by finite products: In [82]:3 implies: A product of two S1(Ω,Γ)-sets need not be an S1(Ω,Γ)-set. In [105]:CH implies the existence of a space with S1(Γ,Γ) whose square does not haveSfin(O,O). In [105]: CH implies existence of a space with S1(O,O) whose squaredoes not have Sfin(O,O).

Problem 13. When is the product of two spaces with property Sfin(A,B) again aspace with Sfin(A,B)? When is the product of two spaces with property S1(A,B)again a space with S1(A,B)?

Results from [39] indicate that game-theoretic conditions yield positive producttheorems.

6.1. Dimension functions. Since the introduction of notions of infinite dimen-sionality there have been proposals for extending the classical Lebesgue coveringdimension function to an ordinal function that in the finite dimensional case as-signs the correct ordinal number to the space and which for each infinite dimen-sional space assigns an ordinal number as measure of the dimensions of the space.One of the ideals of such an extension of the Lebesgue covering dimension functionfrom the finite dimensional spaces to all spaces is that the basic properties of thefinite dimensional dimension function have recognizable analogues for the infinitedimensional dimension functions, such as that the dimension of a subspace does notexceed the dimension of a super space, and that certain relations hold between thedimension function’s value at a product, and its values at the factor spaces. Thereare currently several examples of such dimension functions, each with strong moti-vations. The earliest examples are the so-called small inductive dimension ind(X)and large inductive dimension Ind(X). These dimensions apply to certain count-able dimensional spaces. Further dimension functions that apply more widely toweakly infinite dimensional spaces that are not necessarily countable dimensionalwere introduced by R. Pol [137] and by P. Borst [51] and [52]: Their dimensionfunctions are based on deep descriptive set theoretic ideas. Examples of dimen-sion functions defined for all separable metric spaces are given by L. Babinkostovain [29], [35] and [36] and are based on ideas regarding infinite games. Also V.V.Fedorchuk introduced certain dimension functions in [76] and [77].


To describe the Pol-Borst dimension functions, we consider compact separa-ble metric spaces. Such a space is weakly infinite dimensional if each ω-sequence((Cn, Dn) : n < ω) where each term is a disjoint pair of closed nonempty sets isan inessential sequence. Compactness of the underlying space implies that certainfinite subsequences of such an ω-sequence are already inessential. For compact sep-arable metric spaces there are certain “canonical” such ω-sequences: A sequence((Cn, Dn) : n < ω) is said to be a separating sequence if there is for each pair(C,D) of disjoint closed nonempty sets infinitely many n < ω such that C ⊂ Cn

and D ⊂ Dn. For a separating sequence S = ((Cn, Dn) : n < ω) consider the set

MS = F ⊂ ω : F finite and ((Cn, Dn) : n ∈ F ) is essential.

For finite subsets F and G of ω define F ≺ G if there is a least n ∈ F \ G withF ∩ 0, · · · , n− 1 = G ∩ 0, · · · , n− 1.

Then the underlying space compact separable metric space X is weakly infinitedimensional if, and only if, (MS ,≺) is well-ordered. The ordinal number isomorphicto this well-ordered set depends on the spaceX and is independent of the separatingsequence used. Pol defines this ordinal to be index(X). Using similar considerations,Borst defines an ordinal that will be denoted dimB(X), and proves that for compactseparable metric spaces for which these quantities are defined,

index(X) = ωdimB(X)

(here, the exponentiation is ordinal exponentiation). If X is a compact sub-set of the weakly infinite dimensional compact separable metric space Y , thendimB(X)≤dimB(Y). Borst’s definitions do not presuppose that the spaces in ques-tion are compact, and he proves that

Theorem 6.11 (Borst, [51]). For a separable metric space X, the following areequivalent:

(1) dimB(X) is defined.(2) Each inessential sequence ((Cn, Dn) : n < ω) consisting of pairs of disjoint

closed nonempty sets some finite subsequence is inessential.

In the following result of Borst, C denotes the Cantor set :

Theorem 6.12 (Borst, [51]). (1) For X a finite dimensional metric space theLebesgue covering dimension of the space is equal to dimB(X).

(2) If Y is a closed subspace of the space X and dimB(X) is defined, thendimB(Y ) ≤ dimB(X).

(3) If X is a space for which Ind(X) exists, then dimB(X) ≤ Ind(X). Thisinequality may be strict.

(4) If X is compact, then dimB(X) exists if, and only if, X is weakly infinitedimensional.

(5) If X is locally compact and dimB(X) exists, then dimB(X) = dimB(X×C).

To describe the two dimension functions introduced by Babinkostova we considerthe infinite game of the form Gα

c (A,B) where α is an ordinal and A and B arefamilies of sets. This game between players ONE and TWO is played as follows:In inning γ < α ONE first chooses an Oγ ∈ A, and then TWO responds with a Tγ

where Tγ is a pairwise disjoint family of sets and Tγ refines Oγ . A play

O0, T0, · · · , Oγ , Tγ , · · · (γ < α)


is won by TWO if⋃

γ<α Tγ is a member of B; otherwise, ONE wins the play.Define the game dimension of X to be the ordinal dimG(X), where

1 + dimG(X) = minα : TWO has a winning strategy in Gαc (O,O).

Theorem 6.13 (Babinkostova, [35]). For a separable metric space X(1) The Lebesgue covering dimension of X is n if, and only if, dimG(X) = n.(2) X is countable dimensional if, and only if, dimG(X) = ω.(3) If X is Pol’s space, then dimG(X) = ω + 1.(4) If Y is a closed subspace of X, then dimG(Y ) ≤ dimG(X).(5) dimG(X) ≤ ω1.

The second dimension function is inspired by the classical theorem of Menger andNobeling: Every separable metric spaceX of covering dimension n is homeomorphicto a subspace of R2n+1. But R2n+1 has a natural group structure under which it isa topological group. If (G, ∗) is a topological group and U is an open neighborhoodof the identity element, then

O(U) = x ∗ U : x ∈ G

is an open cover of G. We define

Onbd = O(U) : U an open neighborhood of the identity of G.

Then Onbd is a family of open covers of G.For X a subset of the topological group (G, ∗) we define the neighborhood game

dimension of X to be the ordinal dimnbd(X), where

1 + dimnbd(X) = minα : TWO has a winning strategy in Gαc (Onbd,OX).

Note that in the definition of dimnbd(X), the ambient group (G, ∗) of which X isa subspace may have some influence on the ordinal number dimnbd(X), and thusshould appear as a parameter in this quantity. Where it is clear from context whichgroup is intended, we will neglect this parameter. There are instances where thenumber is independent of the group that X is embedded in:

Theorem 6.14 (Babinkostova, [36]). Let (G, ∗) be a group and let the separablemetric spaces X and Y be subspaces of G.

(1) For each n < ω, dimnbd(X) = n if, and only if, the covering dimension ofX is n.

(2) dimnbd(X) = ω if, and only if, X is countable (but not finite) dimensional.(3) If X ⊆ Y , then dimnbd(X) ≤ dimnbd(Y ) ≤ dimnbd(G)(4) dimnbd(X) ≤ dimG(X).

Thus, for finite or countable dimensional metric spaces it is not important whichambient topological group structure is used in computing the neighborhood gamedimension of the space. By a theorem of Banach and Mazur there is for each sep-arable metric space (X, d) an isometric embedding into C([0, 1]) endowed with themetric derived from the supremum norm. Now (C([0, 1]),+) is a topological group.Thus, a dimension can be defined for (X, d) by computing the neighborhood dimen-sion of the isometric copy in the topological group (C([0, 1]),+). Theorem 6.14 (4)gives an upper bound on the possible values of neighborhood game dimension fora separable metrizable space: The neighborhood game dimension of any isometriccopy of the space in (C([0, 1]),+) does not exceed the game dimension.


Examples in [36] illustrate that the neighborhood game dimension can depend onwhich isometric copy of a space is considered. To illustrate: Consider the stronglyinfinite dimensional totally disconnected complete metric space M of [147]. By aclassical theorem of Lelek there is a metrizable compactification K18 of M for whichthe remainder K \ M is a union of countably many compact finite dimensionalspaces. It is shown in [35] that dimG(K) = ω + 1, while dimG(M) = ω1. Theorem6.14 implies that for each metric d inherited from K the isometric copy of (M, d)in C([0, 1]) will have neighborhood game dimension ω+ 1. But since M is stronglyinfinite dimensional it also has a compatible metric d∗ such that the isometric copyof the metric space (M, d∗) in C([0, 1]) does not have the property Sc(Onbd,OM),from which it follows that the neighborhood game dimension of the correspondingisometric copy of M is ω1. These phenomena indicate that one way to define anunambiguous neighborhood game dimension for a separable metric space might beto take the minimum over the neighborhood game dimensions of isometric copiesof the space in the group (C([0, 1]),+).

Although the game dimension of Pol’s space K is known to be ω + 1, it is notknown what the value of dimB(K) is equal to. Moreover, the relation between theneighborhood game dimension of a compact weakly infinite dimensional space andits Pol-Borst dimension is also not known. Examples given in [36] suggest that theminimum of the neighborhood game dimensions of isometric copies of such a spacein C([0, 1]) may be no larger than the Pol-Borst dimension.

7. What as been left out?

We have omitted surveying a large body of work simply because of the enormityof the task and the necessity of completing an “opening” survey of the area in areasonable time. We have included what we were most ready to write about.

The large emerging body of important work that we did not cover is left foranother time. There is a growing literature on how selection principles “transfer”between a space and the several hyperspaces that have been studied in topology.Readers could get a glimpse of this work in some references like [69] and [70]. Wehave also not featured the fast growing literature on selection principles for weakercovering properties and a corresponding duality theory. Readers interested in thesedevelopments can get a glimpse from references like [38], [67], [107] and [128]. Ex-cept for a small excursion in the section on dimension theory we have also notfeatured the research into the relation between selection principles and increasedlength of the corresponding natural games. It appears that these studies are espe-cially important for their relevance to the theory of products and the preservationof properties in generic extensions. For a glimpse of work done in this subfieldreaders could consult for example [3], [12], [39], [40], [47], [68], [106] and [164].

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Index

Ind(X), 32

X(α), 13

Γ, 10

Γx, 16

Λ, 10

Ω, 6

Ωgp, 11

Ωx, 15

Sc(A,B), 3

Sc(O,O), 3

Sc(O2,O), 3

C(κ), 9

δ(X), 22

Gαc (A,B), 33

Gαfin(A,B), 23

Gα1 (A,B), 4

D, 22

O, 2

O(I), 5

Ogp, 10

On, 3

OXY , 5

Onbd, 34

Onbd, 5

π(X), 22

ψ(X), 8

Sfin(A,B), 2

Sfin(O,O), 2

S1(Λ,Ogp), 10

S1(Ωx,Ωx), 15

S1(A,B), 2

S1(D,D), 22

S1(O,O), 2

S1(Onbd,OGX), 5

LimAn, 17

d(X), 22

ind(X), 32

C, 33

K, 29

M, 35

P, 25

H, 29

A → dBenk , 23

A → (B)nk , 4

E(A,B), 7

Fn(κ, ω), 9

L(X), 8

dimB(X), 33

dimG(X), 34

dimnbd(X), 34

index(X), 33

Cp(X), 9

PR(X), 11

card(X), 14

ht(X), 14

wd(X), 14

ℵ0-bounded, 5

BC, 4

BCκ, 5

cardinal sequence, 14

CH, 4

Corson compact, 13

countable dimensional, 3, 29

countable fan tightness at x, 28

countable strong fan tightness at x, 15

countable tightness, 15

countably fan tight, 28

countably strong fan tight, 16

countably tight at x, 15

density, 22

Eberlein compact, 12

Ellentuck topology, 7

equal convergence, 20

essential sequence, 29

Frechet-Urysohn at x, 17

game dimension, 34

γ-cover, 10

groupable cover, 10

groupable ω-cover, 11

Haver property, 30

height, 14

Hurewicz Conjecture, 26

Hurewicz property, 26

inessential, 29

isolated point, 11

large cover, 10

large inductive dimension, 32

(left) Rothberger bounded, 5

meager, 7

Menger basis property, 2

Menger property, 2

Menger’s Conjecture, 24

neighborhood assignment, 25

neighborhood game dimension, 34

nowhere dense, 7

ω-cover, 6

ω-shrinkable, 18

ordinary partition symbol, 4

π-base, 22

partition, 29

Pixley-Roy hyperspace, 12

Pixley-Roy topology, 11

productively Lindelof, 25

43


property(αi), 18(α3/2), 20

(α0), 19(δ), 17(USC)s, 21

pseudo-character, 8Pytkeev property, 18

quasinormal convergence, 20

R-separability, 22Radon-Nikodym compact, 12Reznichenko property, 17Rothber bounded, 5Rothberger property, 2

scattered, 11screenable, 3selective screenability, 3separating sequence, 33sequence selection property, 19small inductive dimension, 32space

D-space, 25δ-space, 17ε-space, 6Frechet-Urysohn, 17Gerlits-Nagy, 10Hurewicz, 26indestructibly Lindelof, 8κ-Frechet-Urysohn, 17Menger, 2, 24Michael, 25M-separable, 28P-space, 11QN∗-space, 21QN-space, 20QN∗-space, 21Rothberger, 2, 6R-separable, 22selectively separable, 28strictly Frechet-Urysohn, 17weakly Frechet-Urysohn, 17wQN∗-space, 21wQN-space, 20wQN∗-space, 21

strictly Frechet-Urysohn at x, 17strong Eberlein compact, 13strong measure zero, 1, 5strongly infinite dimensional, 3, 29

The Borel Conjecture, 4The Michael Space Problem, 25the property of Baire, 7

van Douwen’s problem, 25

weakly infinite dimensional, 3, 29width, 14

THE COMBINATORICS OF OPEN COVERS

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