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SOME NEW DIRECTIONS IN

INFINITE-COMBINATORIAL TOPOLOGY

BOAZ TSABAN

Abstract. We give a light introduction to selection principlesin topology, a young subfield of infinite-combinatorial topology.Emphasis is put on the modern approach to the problems it dealswith. Recent results are described, and open problems are stated.Some results which do not appear elsewhere are also included, withproofs.

Contents

0. Introduction 21. The Menger-Hurewicz conjectures 31.1. The Menger and Hurewicz properties 31.2. Consistent counter examples 41.3. Counter examples in ZFC 52. The Borel Conjecture 62.1. Strong measure zero 62.2. Rothberger’s property 83. Classification 93.1. More properties 93.2. ω-covers 93.3. Arkhangel’skii’s property 93.4. The Scheepers Diagram 103.5. Borel covers 113.6. Borel’s Conjecture revisited 114. Preservation of properties 134.1. Additivity 134.2. Hereditarity 145. The Minimal Tower Problem and τ -covers 155.1. “Rich” covers of spaces 155.2. The Minimal Tower Problem 15

Partially supported by the Golda Meir Fund and the Edmund Landau Center forResearch in Mathematical Analysis and Related Areas, sponsored by the MinervaFoundation (Germany).

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http://arXiv.org/abs/math/0409069v1

2 BOAZ TSABAN

5.3. A dictionary 165.4. Topological approximations of the Minimal Tower Problem 165.5. Tougher topological approximations 175.6. Known implications and critical cardinalities 185.7. A table of open problems 215.8. The Minimal Tower Problem revisited 216. Some connections with other fields 226.1. Ramsey theory 226.2. Countably distinct representatives and splittability 246.3. An additivity problem 256.4. Arkhangel’skii duality theory 267. Conclusions 277.1. Acknowledgments 27References 27

0. Introduction

The modern era of what we call infinite combinatorial topology , orselection principles in mathematics began with Scheepers’ paper [36]and the subsequent work [20]. In these works, a unified frameworkwas given that extends many particular investigations carried in theclassical era. The current paper aims to give the reader a taste of thefield by telling six stories, each shedding light on one specific theme.The stories are short, but in a sense never ending, since each of themposes several open problems, and more are expected to arise when theseare solved.

This is not intended to be a systematic exposition to the field, noteven when we limit our scope to selection principles in topology. Forthat see Scheepers’ survey [40] as well as Kocinac’s [21]. Rather, wedescribe themes and results with which we are familiar. This impliesthe disadvantage that we are often quoting our own results, which onlyform a tiny portion of the field.1

Some open problems are presented here. For many more see [50]. Af-ter reading this introduction, the reader can proceed directly to some ofthe works of the experts in the filed (or in closely related fields), suchas:2 Liljana Babinkostova, Taras Banakh, Tomek Bartoszynski, Lev

1To partially compensate for this, the name of the present author is never ex-plicitly mentioned in the paper.

2This very incomplete list is ordered alphabetically. We did not give referencesto works not explicitly mentioned in this paper, but the reader can find some ofthese in the bibliographies of the given references, most notably, in [40, 21]. We

SOME NEW DIRECTIONS IN INFINITE-COMBINATORIAL TOPOLOGY 3

Bukovski, Krzysztof Ciesielski, David Fremlin, Fred Galvin, SalvadorGarcia-Ferreira, Janos Gerlits, Cosimo Guido, Istvan Juhasz, LjubisaKocinac, Adam Krawczyk, Henryk Michalewski, Arnold Miller, Zsig-mond Nagy, Andrzej Nowik, Janusz Pawlikowski, Roman Pol, IreneuszRec law, Miroslav Repicky, Masami Sakai, Marion Scheepers, LajosSoukup, Paul Szeptycki, Stevo Todorcevic, Tomasz Weiss, LubomyrZdomsky, and many others.

Notation. In most cases, the notation and terminology we use is Scheep-ers’ modern one, and we do not pay special attention to the historicalpredecessors of the notations we use. The reader can use the index atthe end of the paper to locate the definitions he is missing.

By set of reals we usually mean a zero-dimensional, separable metriz-able space, though some of the results hold in more general situations.

Apology. Some of the results might be miscredited or misquoted (orboth). Please let us know of any mistake you find and we will correctit in the online version of this paper [53].

1. The Menger-Hurewicz conjectures

1.1. The Menger and Hurewicz properties. In 1924, Menger [26]introduced the following basis covering property for a metric space X:

For each basis B of X, there exists a sequence Bnn∈N

in B such that limn→∞ diam(Bn) = 0 and X =⋃

n Bn.

It is an amusing exercise to show that every compact, and even σ-compact space has this property. Menger conjectured that this prop-erty characterizes σ-compactness. In 1925, Hurewicz [19] introducedtwo properties of the following prototype. For collections A,B of coversof a space X, define

Ufin(A,B): For each sequence Unn∈N of members of A which do not con-tain a finite subcover, there exist finite (possibly empty) subsetsFn ⊆ Un, n ∈ N, such that ∪Fnn∈N ∈ B.

Hurewicz proved that for O the collection of all open covers of X,Ufin(O,O) is equivalent to Menger’s basis property. Hurewicz did notsettle Menger’s conjecture, but he suggested a more modest one: Callan open cover U of X a γ-cover if U is infinite, and each x ∈ X belongs

should also comment that not all works of the authors are formulated using thesystematic notation of Scheepers which we use here, and probably some of thementioned experts do not consider themselves as working in the discussed field(but undoubtly, each of them made significant contributions to the field).

4 BOAZ TSABAN

AAA A

B

Figure 1. Ufin(A,B)

to all but finitely many members of U . Let Γ denote the collection ofall open γ-covers of X. Clearly,

σ-compact ⇒ Ufin(O, Γ) ⇒ Ufin(O,O),

and Hurewicz conjectured that for metric spaces, Ufin(O, Γ) (now knownas the Hurewicz property) characterizes σ-compactness.

1.2. Consistent counter examples. Apparently, it did not take verylong to find out that these conjectures are false assuming the Contin-uum Hypothesis: Observe that every uncountable Fσ set of reals con-tains an uncountable perfect set, which in turn contains an uncountableset which is both meager (i.e., of Baire first category) and null (i.e.,of Lebesgue measure zero). A set of reals L is a Luzin set if it isuncountable, but for each meager set M , L ∩ M is countable. It wasproved by Mahlo (1913) and Luzin (1914) that the Continuum Hypoth-esis implies the existence of a Luzin set. Sierpinski [43] (1928) pointedout that each Luzin set has Menger’s property Ufin(O,O) (hint: If wecover a countable dense subset of the Luzin set by open sets, then theuncovered part is meager and therefore countable), and is therefore acounter example to Menger’s conjecture.

Similarly, a set of reals S is a Sierpinski set if it is uncountable, butfor each null set N , S ∩ N is countable. Sierpinski showed that theContinuum Hypothesis implies the existence of such sets, and it can beshown that Sierpinski sets have the Hurewicz property, and is thereforea counter-examples to Hurewicz’ (and therefore Menger’s) conjecture.


Why must a Sierpinski set satisfy Hurewicz’ property Ufin(O, Γ)?A classical proof can be carried using Egoroff’s Theorem, but let ussee how a modern, combinatorial proof goes [41]. The Baire spaceNN (a Tychonoff power of the discrete space N) carries an interestingcombinatorial structure: For f, g ∈ NN, write

f ≤∗ g if f(n) ≤ g(n) for all but finitely many n.

B ⊆ NN is bounded if there exists g ∈ NN such that f ≤∗ g for all f ∈ B.D ⊆ NN is dominating if for each g ∈ NN there exists f ∈ D such thatg ≤∗ f . It is easy to see that a countable union of bounded sets in NN

is bounded, and that compact (and therefore σ-compact) subsets of NN

are bounded. The following theorem is usually attributed to Hurewicz,but an explicit proof of it was first given in Re law [31].

Theorem 1.1 (Hurewicz). For a set of reals X:

(1) X satisfies Ufin(O, Γ) if, and only if, all continuous images ofX in NN are bounded.

(2) X satisfies Ufin(O,O) if, and only if, all continuous images ofX in NN are not dominating.

Assume that S ⊆ [0, 1] is a Sierpinski set and Ψ : S → NN is contin-uous. Then Ψ can be extended to all of [0, 1] as a Borel function. Bya theorem of Luzin, there exists for each n a closed subset Cn of [0, 1]such that µ(Cn) ≥ 1−1/n, and such that Ψ is continuous on Cn. SinceCn is compact, Φ[Cn] is bounded in NN. The set N = [0, 1] \

⋃

n Cn isnull, and so its intersection with S is countable. Consequently, Ψ[S]is contained in a union of countably many bounded sets in NN, and istherefore bounded.

1.3. Counter examples in ZFC. But are the conjectures consistent?It turns out that the answer is negative. Surprisingly, it was onlyrecently that this question was clarified, again using a combinatorialapproach. Let b denote the minimal size of an unbounded subset ofNN, and d denote the minimal size of a dominating subset of NN. Thecritical cardinality of a (nontrivial) collection J of sets of reals is

non(J ) = min|X| : X ⊆ R, X 6∈ J .

By Hurewicz’ Theorem 1.1, non(Ufin(O, Γ)) = b, and non(Ufin(O,O))= d. In 1988, Fremlin and Miller [15] used their celebrated dichotomicargument to refute Menger’s conjecture (in ZFC): By the last observa-tion, if ℵ1 < d then any set of reals of size ℵ1 will do; and if ℵ1 = d,then one can use a sophisticated combinatorial construction. This di-chotomic argument was extended in 1996 to refute Hurewicz’ conjecture

6 BOAZ TSABAN

[20], this time using the dichotomy ℵ1 < b or ℵ1 = b and even moresophisticated combinatorial arguments in the second case.

A simple construction was very recently found (in [5] and extendedin [7]) to refute both conjectures, and not on a dichotomic basis: Thereexists a non σ-compact set of reals H of size b which has the Hurewiczproperty. The construction does not use any special hypothesis: LetN∪∞ be the one point compactification of N, and Z be the functionsf ∈ N(N ∪ ∞) such that for each n, f(n) ≤ f(n + 1), and strictinequality holds as long as f(n) < ∞. For a finite increasing sequences of natural numbers, let qs be the element of Z extending s and beingequal to ∞ on all new n’s. Then the collection Q of all these elementsqs is dense in Z. Define a b-scale to be an unbounded set fα : α <b ⊆ NN of increasing functions, such that fα ≤∗ fβ whenever α < β.It is an easy exercise to construct a b-scale in ZFC.

Theorem 1.2 (Bartoszynski, et. al. [7]). Let H be a union of a b-scaleand Q. Then all finite powers of H satisfy Ufin(O, Γ) (but H is notσ-compact).

Consequently, there exists a counter example GH to the Hurewiczconjecture, such that |GH | = b and GH is a subgroup of R [49] (cf.[12]).

Similarly, it was shown in [7] that there exists a counter example ofsize d to the Menger conjecture. However it is open whether the grouptheoretic version also holds.

Problem 1.3. Does there exist (in ZFC) a subgroup GM of R suchthat |GM | = d and GM has Menger’s property Ufin(O,O)?

Zdomsky (personal communication) claims that the construction ofChaber and Pol [12] can probably be modified to obtain a positivesolution to Problem 1.3, at least in the case that d is regular.

2. The Borel Conjecture

2.1. Strong measure zero. Recall that a set of reals X is null if foreach positive ǫ there exists a cover Inn∈N of X such that

∑

n diam(In) <ǫ. In his 1919 paper [10], Borel introduced the following stronger prop-erty: A set of reals X is strongly null (or: has strong measure zero) if,for each sequence ǫnn∈N of positive reals, there exists a cover Inn∈N

of X such that diam(In) < ǫn for all n. But Borel was unable toconstruct a nontrivial (that is, an uncountable) example of a stronglynull set. He therefore conjectured that there exist no such examples.Sierpinski (1928) observed that every Luzin set is strongly null (see thehint on page 4 for the reason), thus the Continuum Hypothesis implies


that Borel’s Conjecture is false. Sierpinski asked whether the propertyof being strongly null is preserved under taking homeomorphic (or evencontinuous) images. The answer, given by Rothberger (1941) in [32],is negative under the Continuum Hypothesis.

If we carefully check Rothberger’s argument, we can obtain a slightlystronger result without making the proof more complicated, and withthe benefit of understanding the underlying combinatorics better. The-orem 2.1 and Proposition 2.2 below are probably folklore, but we do notknow of a satisfactory reference for them so we give complete proofs.(The proof of Theorem 2.1 is a modification of the proof of [4, The-orem 2.9].) Let SMZ denote the collection of strongly null sets ofreals. A subset A of NN is strongly unbounded if for each f ∈ NN,|g ∈ A : g ≤∗ f| < |A|. Observe that there exist strongly unboundedsets of sizes b and d, thus any of the hypotheses non(SMZ) = b ornon(SMZ) = d (in particular, the Continuum Hypothesis) implies theassumption in the following theorem.

Theorem 2.1. Assume that there exists a strongly unbounded set ofsize non(SMZ). Then there exist a strongly null set of reals X and acontinuous image Y of X such that Y is not strongly null.

Proof. Let κ = non(SMZ). Then there exist: A strongly unboundedset A = fα : α < κ, and a set of reals Y = yα : α < κ that isnot strongly null. By standard translation arguments (see, e.g., [52])we may assume that Y ⊆ [0, 1] and therefore think of Y as a subset of0, 1N (0, 1N is Cantor’s space, which is endowed with the producttopology). Consequently, the set A′ = f ′

α : α < κ, where for eachα, f ′

α(n) = 2fα(n) + yα(n) for all n, is also strongly unbounded, andthe mapping A′ → Y defined by f(n) 7→ f(n) mod 2 is continuous andsurjective.

It remains to show that A′ is a continuous image of a strongly nullset of reals X. Let Ψ : NN → R \ Q be a homeomorphism (e.g.,taking continued fractions), and let X = Ψ[A′]. We claim that X isstrongly null. Indeed, assume that ǫnn∈N is a sequence of positivereals. Enumerate Q = qn : n ∈ N, and choose for each n an openinterval I2n of length less than ǫ2n such that qn ∈ I2n. Let U =

⋃

n I2n.Then R \ U is σ-compact, thus B = Ψ−1[R \ U ] = NN \ Ψ−1[U ] isa σ-compact and therefore bounded subset of NN. As A′ is stronglyunbounded, |A′\Ψ−1[U ]| = |A′∩B| < κ = non(SMZ), thus Ψ[A′∩B] =Ψ[A′] \ U is strongly null, so we can find intervals I2n+1 of diameter atmost ǫ2n+1 covering this set, and notice that

X = Ψ[A′] ⊆ (Ψ[A′] \ U) ∪ U ⊆⋃

n∈N

In.

8 BOAZ TSABAN

From Theorem 2.1 it is possible to deduce that SMZ is not provablyclosed under taking homeomorphic images. The argument in the fol-lowing proof, that is probably similar to Rothberger’s, was pointed outto us by T. Weiss. Observe that SMZ is hereditary (that is, if X isstrongly null and Y is a subset of X, then Y is strongly null too), andthat it is preserved under taking uniformly continuous images.

Proposition 2.2. If a hereditary property P is not preserved undertaking continuous images, but is preserved under taking uniformly con-tinuous images, then it is not preserved under taking homeomorphicimages.

Proof. Assume that X satisfies P , Y does not satisfy P , and f : X → Yis a continuous surjection. Let X ⊆ X (so that X satisfies P ) be suchthat f : X → Y is a (continuous) bijection. Then the set f ⊆ X × Y

(we identify f with its graph) is homeomorphic to X which satisfies P ,but the projection of f on the second coordinate, which is a uniformlycontinuous image of f , is equal to Y . Thus f does not satisfy P .

2.2. Rothberger’s property. This lead Rothberger to introduce thefollowing topological version of strong measure zero (which is preservedunder taking continuous images). Again, let A and B be collections ofopen covers of a topological space X. Consider the following prototypeof a selection hypothesis.

S1(A,B): For each sequence Unn∈N of members of A, there exist mem-bers Un ∈ Un, n ∈ N, such that Unn∈N ∈ B.

AA A

B

A

Figure 2. S1(A,B)


Then Rothberger introduced the case A = B = O (the collectionof all open covers).3 Clearly, Rothberger’s property S1(O,O) impliesbeing strongly null, and the usual argument shows that every Luzinset L satisfies S1(O,O). Moreover, Fremlin and Miller [15] proved thatfor a metric space 〈X, d〉, S1(O,O) is the same as having strong measurezero with respect to all metrics which generate the same topology asthe one defined by d.

The question of the consistency of Borel’s Conjecture was settled in1976, when Laver in his deep work [23] showed that Borel’s Conjectureis consistent. We will return to Borel’s Conjecture in Section 3.6.

3. Classification

3.1. More properties. Having the terminology introduced thus far,we can also consider the properties S1(Γ,O), S1(Γ, Γ), and S1(O, Γ).The last property turns out trivial (consider an open cover with noγ-subcover), but the first two make sense even in the restricted settingof sets of reals. These properties turn out much more restrictive thanMenger’s property Ufin(O,O), but they do not admit an analogue ofthe Borel conjecture. In fact, the set H from Theorem 1.2 satisfiesS1(Γ,O) (by an argument similar to that in Theorem 2.1) [7], and infact there always exist uncountable elements satisfying S1(Γ, Γ) [20, 39].

Problem 3.1. Does the set H from Theorem 1.2 satisfy S1(Γ, Γ)? Doesthere always exist a set of size b satisfying S1(Γ, Γ)?

3.2. ω-covers. We need not stop here, and may wish to consider otherimportant types of covers which appeared in the literature. An opencover U of X is an ω-cover of X if no single member of U covers X,but for each finite F ⊆ X there exists a single member of U coveringF . Let Ω denote the collection of open ω-covers of X. Then S1(Ω, Γ) isequivalent to the γ-property introduced by Gerlits and Nagy (1982) in[17], and S1(Ω, Ω) was studied by Sakai (1988) in [34], both propertiesnaturally arising in the study of the space of continuous real valuedfunctions on X (we will return to this in Section 6.4).

3.3. Arkhangel’skii’s property. Another prototype for a selectionhypothesis generalizes a property studied by Arkhangel’skii, also inthe context of function spaces, in 1986 [1]. The prototype is similar toUfin(A,B), but we do not “glue” the finite subcollections.

3Traditionally, Rothberger denoted this property by C′′. The reason was thatsome denoted strong measure zero by C, so Rothberger used C′ to denote continuousimages of elements of C, and C′′ to be what we now call S1(O,O), since it impliesC′.

10 BOAZ TSABAN

Sfin(A,B): For each sequence Unn∈N of members of A, there exist finite(possibly empty) subsets Fn ⊆ Un, n ∈ N, such that

⋃

n∈NFn ∈

B.

Then the property studied by Arkhangel’skii is equivalent to Sfin(Ω, Ω).

3.4. The Scheepers Diagram. Thus far we have a selection hypothe-sis corresponding to each member of the 27 element set S1, Sfin, Ufin×Γ, Ω,O2. Fortunately, it suffices to consider only some of them. First,observe that in the cases we consider,

S1(A,B) ⇒ Sfin(A,B) ⇒ Ufin(A,B),

and we have the following monotonicity property: For Π ∈ S1, Sfin, Ufin,if A ⊆ C and B ⊆ D, then:

Π(A,B) → Π(A,D)↑ ↑

Π(C,B) → Π(C,D)

After removing trivial properties and proving equivalences among theremaining ones (see [20] for a summary of these), we get the ScheepersDiagram (Figure 3). In this diagram, as in the ones to follow, anarrow denotes implication, and below each property we wrote its criticalcardinality. (cov(M) denotes the minimal cardinality of a cover of R

by meager sets, and p is the pseudo-intersection number to be definedin Section 5.2. See [9] for information on these cardinals as well asother cardinals which we mention later.)

Ufin(Γ, Γ)b

// Ufin(Γ,Ω)d

// Ufin(Γ,O)d

Sfin(Γ, Ω)d

66llllll

S1(Γ, Γ)b

//

66mmmmmmmmmmmmmmmmmmmm

S1(Γ, Ω)d

//

66mmmmmm

S1(Γ,O)d

88q

qq

qq

qq

qq

qq

qq

qq

Sfin(Ω, Ω)d

OO

S1(Ω, Γ)p

//

OO

S1(Ω, Ω)cov(M)

OO

//

66nnnnnn

S1(O,O)cov(M)

OO

Figure 3. The Scheepers Diagram

There remain only two problems concerning this diagram.

Problem 3.2.


(1) Does Ufin(Γ, Ω) imply Sfin(Γ, Ω)?(2) If not, does Ufin(Γ, Γ) imply Sfin(Γ, Ω)?

All other implications are settled in [36, 20], using two methods. Oneapproach uses consistency results concerning the values of the criticalcardinalities. For example, it is consistent that b < d, thus none ofthe properties with critical cardinality d can imply any of those withcritical cardinality b. Another approach is by transfinite constructionsunder the Continuum Hypothesis (such as special kinds of Luzin andSierpinski sets). Recently, an approach combining these two approacheswas investigated – see, e.g., [11, 3, 14].

3.5. Borel covers. There are other natural types of covers which ap-pear in the literature, but probably the first natural question is: Whathappens if we replace “open” by “countable Borel” in the types of cov-ers which we consider? Let B,BΩ,BΓ denote the collections of countableBorel covers, ω-covers, and γ-covers of the given space, respectively.4

It turns out that the same analysis is applicable when we plug in thesefamilies instead of O, Ω, Γ, and in fact one gets more equivalences.Moreover, some of the resulting properties turn out equivalent to prop-erties which appeared in the literature in other guises. This is shownin [41], where it is also shown that no arrows can be added (exceptperhaps those corresponding to Problem 3.2) to the extended diagram(Figure 4).

3.6. Borel’s Conjecture revisited. It is easy to verify that everycountable set of reals X satisfies the strongest property in the ex-tended Scheepers Diagram 4, namely, S1(BΩ,BΓ). By Laver’s resultmentioned in Section 2.2, we know that it is consistent that all proper-ties between S1(BΩ,BΓ) and S1(O,O) (inclusive) are consistent to holdonly for countable sets of reals. Recall that all other classes in theoriginal Scheepers Diagram 3 contain uncountable elements (Section1.3), in other words, they cannot satisfy an analogue of Borel’s Conjec-ture. However, by a result of Miller, Borel’s Conjecture for S1(BΓ,BΓ)is consistent [7].

Problem 3.3. Is it consistent that every set of reals which satisfiesS1(BΓ,B) is countable? What about Sfin(BΩ,BΩ) and S1(BΓ,BΩ)?

4We abuse the notation by using B sometimes for a general family of coversof a space, and sometimes for the family of countable Borel covers of the space.However, our meaning should be clear from the context.

12 BOAZ TSABAN

Ufin(Γ, Γ)b

//?

%%

Ufin(Γ, Ω)d

//?

Ufin(Γ,O)d

Sfin(Γ, Ω)d

99sss

S1(Γ, Γ)b

//

99r

rr

rr

rr

rr

rr

rr

r

S1(Γ, Ω)d

//

99sss

S1(Γ,O)d

==z

zz

zz

zz

zz

zz

S1(BΓ,BΓ)b

::uuu

// S1(BΓ,BΩ)d

88qqq

// S1(BΓ,B)d

::uuu

Sfin(Ω, Ω)d

OO

Sfin(BΩ,BΩ)d

OO

66mmmmmmmmmm

S1(Ω, Γ)p

//

OO

S1(Ω, Ω)cov(M)

OO

//

EE

S1(O,O)cov(M)

OO

S1(BΩ,BΓ)p

OO

//

;;www

S1(BΩ,BΩ)cov(M)

OO

99sss

// S1(B,B)cov(M)

OO

;;www

Figure 4. The extended Scheepers Diagram

A combinatorial formulation of the first question in Problem 3.3 isobtained by replacing S1(BΓ,BΩ) with the equivalent property “everyBorel image in NN is not dominating”.

Other interesting investigations in this direction are of the form: Is itconsistent that a certain property in the diagram satisfies Borel’s Con-jecture, whereas another one does not? Some results in this directionare the following.

Theorem 3.4 (Miller [30]).

(1) Borel’s Conjecture for S1(O,O) implies Borel’s Conjecture (forstrong measure zero);

(2) Borel’s Conjecture for S1(Ω, Γ) does not imply Borel’s Conjec-ture.

The proofs use, of course, combinatorial arguments (and forcing inthe second case: The model is obtained by adding ℵ2 dominating realswith a finite support iteration to a model of the Continuum Hypothe-sis). Using yet more combinatorial arguments, it is possible to extendTheorem 3.4.

Theorem 3.5 (Weiss, et. al. [52]). Borel’s Conjecture for S1(Ω, Ω) im-plies Borel’s Conjecture. Consequently, Borel’s Conjecture for S1(Ω, Γ)does not imply Borel’s Conjecture for S1(Ω, Ω).


This settles completely this investigation when we restrict attentionto the original Scheepers Diagram 3.

4. Preservation of properties

4.1. Additivity. In [20] (1996), Just, Miller, Scheepers, and Szeptyckiraised the following additivity problem: It is easy to see that some ofthe properties in the Scheepers diagram are (provably) preserved un-der taking finite and even countable unions (i.e., they are σ-additive).What about the remaining ones? Figure 4.2(a) summarizes the knowl-edge that was available at the point the question was posed, where thepositions are according to Figure 3.

X // ? // X

?

;;xxxx

? //

::v

vv

vv

vv

vv

v

? //

::v

vv

v

X

==|

||

||

||

|

?

OO

× //

OO

?

OO

//

::v

vv

v

X

OO

(a)

X // × // X

×

;;x

xx

X //

;;v

vv

vv

vv

vv

v

× //

;;v

vv

X

>>|

||

||

||

|

×

OO

× //

OO

×

OO

//

;;v

vv

X

OO

(b)

Figure 5. The additivity problem (a) and its solution (b)

In 1999, Scheepers [39] proved that the answer is positive for S1(Γ, Γ).The key observation towards solving the problem for the remain-

ing properties was the following analogue of Hurewicz’ Theorem 1.1.According to Blass, a family Y ⊆ NN is finitely dominating if foreach g ∈ NN there exist k and f1, . . . , fk ∈ Y such that g(n) ≤maxf1(n), . . . , fk(n) for all but finitely many n.

Theorem 4.1 ([45]). A set of reals X satisfies Ufin(Γ, Ω) if, and onlyif, each continuous image of X in NN is not finitely dominating.

Motivated by this observation, the following theorem was proved.(As usual, c = 2ℵ0 denotes the size of the continuum.)

Theorem 4.2 (Bartoszynski, Shelah, et. al. [6]). Assume the Contin-uum Hypothesis (or just cov(M) = c). Then There exist sets of realsL0, and L1 satisfying S1(BΩ,BΩ) such that L0 ∪L1 is finitely dominat-ing.

The proof used a tricky “power sharing” between L0 and L1 duringtheir transfinite-inductive construction.

14 BOAZ TSABAN

Consequently, none of the remaining properties is provably preservedunder taking finite unions (Figure 4.2(b)). This also settled all corre-sponding problems in the Borel case (see the extended diagram 4).Interestingly, the simple observation in Theorem 4.1 was also the keybehind proving that consistently (namely, assuming NCF), Ufin(O, Ω)is σ-additive.

Bad transmission of knowledge. The transmission of knowledge con-cerning the additivity problem was very poor (take a deep breathe):It posteriorly turns out that if we restrict attention to the open caseonly, then the additivity problem was already implicitly solved earlier.In 1999, Scheepers [38] constructed sets of reals L0, and L1 satisfyingS1(Ω, Ω) such that L0 + L1 is finitely dominating. It is easy to see thatthis implies that L0 ∪L1 is finitely dominating, which settles the prob-lem if we add the missing ingredient Theorem 4.1, which, funnily, seemsto be the simplest part of the solution. Scheepers was unaware of thisobservation and consequently of that solving the additivity problemcompletely, but he did point out that his construction implied that theproperties between S1(Ω, Ω) and Sfin(Ω, Ω) (inclusive) are not provablyadditive. In turn, we were not aware of this when we proved Theorem4.2. On top of that, Theorem 4.1, the open part of Theorem 4.2, andthe result concerning NCF were independently proved by Eiseworthand Just in [14], and similar results were also independently obtainedby Banakh, Nickolas, and Sanchis in [2]. As if this is not enough, themain ingredient of the result concerning NCF was also independentlyobtained by Blass [8].

This is a good point to recommend the reader announce his newresults in the SPM Bulletin (see [51]), so to avoid similar situations.

4.2. Hereditarity. A property (or a class of topological spaces) ishereditary if it is preserved under taking subsets. Despite the factthat the properties in the Scheepers diagram are (intuitively) notionsof smallness, none of them is (provably) hereditary. Define a topol-ogy on the space P (N) of all sets of natural numbers by identifying itwith 0, 1N. Note that [N]<ℵ0 , the collection of finite subsets of N, isdense in P (N). By taking increasing enumerations, we have that theRothberger space [N]ℵ0 = P (N)\[N]<ℵ0 is identified with the space of in-creasing sequences of natural numbers, which in turn is homeomorphicto the Baire space NN.

Theorem 4.3 (Bartoszynski, et. al. [7]). Assuming (a portion of) theContinuum Hypothesis, there exists a set of reals X satisfying S1(Ω, Γ)


and a countable subset Q of X, such that X \ Q does not satisfyUfin(O,O).

The proof for this assertion is a modification of the construction ofGalvin and Miller [16], with X ⊆ P (N), Q = [N]<ℵ0 , so that X \[N]<ℵ0 ⊆ [N]ℵ0 is dominating (when viewed as a subset of NN) which iswhat we look for in light of Hurewicz’ Theorem 1.1.

However, in the Borel case, S1(B,B) and all properties of the formΠ(BΓ,A) are hereditary. This is immediate from the combinatorialcharacterizations [41], but is also easy to prove directly [7]. However,not all properties in the Borel case are hereditary, e.g., Miller [28]proved that assuming the Continuum Hypothesis, S1(BΩ,BΓ) is nothereditary.

Problem 4.4. Is any of S1(BΩ,BΩ) or Sfin(BΩ,BΩ) hereditary?

5. The Minimal Tower Problem and τ-covers

5.1. “Rich” covers of spaces. 5 To avoid trivialities, let us decidethat when we say that U is a cover of X, we mean that X = ∪U (theusual requirement), and in addition no single member of U covers X.We also always assume that the space X is infinite.U is a large cover of X if for each x ∈ X, there exist infinitely many

U ∈ U such that x ∈ U . Let Λ denote the collection of large opencovers of X. Then

Γ ⊆ Ω ⊆ Λ,

the last implication being a cute exercise.U is a τ -cover of X if it is a large cover of X, and for each x, y ∈ X,

at least one of the sets U ∈ U : x ∈ U and y 6∈ U or U ∈ U : y ∈U and x 6∈ U is finite. This is a reminiscent of the negation of theT1 property, where here the notion is applied “modulo finite”. Let T(uppercase τ) denote the collection of open τ -covers of X. Then

Γ ⊆ T ⊆ Ω.

The motivation behind the definition of τ -covers is the Minimal TowerProblem.

5.2. The Minimal Tower Problem. Recall that [N]ℵ0 is the col-lection of infinite subsets of N. A ⊆∗ B means that A \ B is finite.F ⊆ [N]ℵ0 is centered if for each finite A ⊆ F , ∩A is infinite. A ∈ [N]ℵ0

5Richness can be viewed as some sort of redundancy, and some people are richerthan others. This is precisely the case with rich covers .

16 BOAZ TSABAN

is a pseudo-intersection of F if A ⊆∗ B for all B ∈ F . Let p de-note the minimal size of a centered family F ⊆ [N]ℵ0 with no pseudo-intersection, and t denote the minimal size of a ⊆∗-linearly orderedfamily F ⊆ [N]ℵ0 which has no pseudo-intersection. Then p ≤ t, andthe Minimal Tower Problem is:

Problem 5.1. Is it provable that p = t?

This is one of the major and oldest problems of infinitary combina-torics. Allusions to this problem can be found in Rothberger’s 1940’sworks (see, e.g., [33]). The problem is explicitly mentioned in vanDouwen’s survey [13], and quoted in Vaughan’s survey [54, Problem333], where it is considered the most interesting open problem in thefield. Extensive work of Shelah and others in the field solved all prob-lems of this sort which were mentioned in [13], except for the MinimalTower Problem.

5.3. A dictionary. How is this problem related to τ -covers? Eachtype of rich cover corresponds to some combinatorial notion (of rich-ness), as follows. If we confine attention to sets of reals, then we mayassume that all open covers we consider are countable. Thus, assumethat U = Unn∈N is a countable (not necessarily open) cover of X,and define the Marczewski characteristic function of U [25] by

h(x) = n : x ∈ Un,

so that h : X → P (N). h is continuous if the sets Un are clopen, andBorel if the sets Un are Borel.

We have the following dictionary translating properties of U to prop-erties of the image h[X].

Unn∈N h[X]

large cover subset of [N]ℵ0

ω-cover centeredτ -cover linearly ordered by ⊆∗

γ-cover cofinite setscontains a γ-cover has a pseudo-intersection

(Note that, since we assume that X 6⊆ Un for all n, we have that h[X]is centered if, and only if, it is a base for a nonprincipal filter on N.)

5.4. Topological approximations of the Minimal Tower Prob-

lem. For families B ⊆ A of covers of a space X, define the property Achoose B as follows.

(

AB

)

: For each U ∈ A we can choose a subset V ⊆ U such that V ∈ B.


This is a prototype for many classical topological notions, most notablycompactness and being Lindelof.

In 1982, Gerlits and Nagy [17] introduced the γ-property(

ΩΓ

)

, andproved that it is equivalent to S1(Ω, Γ).

By the dictionary (Section 5.3), we have that

(1) non(

ΩΓ

)

= non(

BΩ

BΓ

)

= p; and

(2) non(

TΓ

)

= non(

BT

BΓ

)

= t.

Clearly,(

ΩΓ

)

⊆(

TΓ

)

, and(

BΩ

BΓ

)

⊆(

BT

BΓ

)

. We therefore have the followingtopological problems related to the Minimal Tower Problem.Problem 5.2.

(1) Is(

ΩΓ

)

=(

TΓ

)

?

(2) Is(

BΩ

BΓ

)

=(

BT

BΓ

)

?

Observe that the answer is “No” for both problems if p < t is con-sistent.

But it turns out that both problems can be solved outright in ZFC:The first problem was solved by Shelah [44], who proved that 0, 1N

satisfies(

TΓ

)

. But a set satisfying(

ΩΓ

)

must be totally imperfect. Amodification of Shelah’s construction gave in [48] a negative solution tothe second question as well: Indeed, NN satisfies

(

TΓ

)

, and Borel images

of NN are analytic, and can therefore be represented as continuousimages of NN, so Borel images of NN satisfy

(

TΓ

)

, and therefore NN

satisfies(

BT

BΓ

)

[48].

5.5. Tougher topological approximations. It is easy to see thatnon(S1(Ω, Γ)) = non(S1(BΩ,BΓ)) = p, and non(S1(T, Γ)) = non(S1(BT,BΓ)) = t [44], and we have the following implications:

S1(Ω, Γ) → S1(T, Γ)↑ ↑

S1(BΩ,BΓ) → S1(BT,BΓ)

We therefore arrive at the following “tighter” approximations.Problem 5.3.

(1) Is S1(Ω, Γ) = S1(T, Γ)?(2) Is S1(BΩ,BΓ) = S1(BT,BΓ)?

Again, p < t implies a negative answer for both problems. It turnsout that the answer is indeed negative, at least assuming the Contin-uum Hypothesis [48]. The main ingredients in the proof of this are thefollowing: S1(BT,BΓ) = S1(BΓ,BΓ) ∩

(

BT

BΓ

)

, and is therefore countablyadditive. Now, the Continuum Hypothesis implies that there exists a

18 BOAZ TSABAN

hereditary S1(BΩ,BΓ)-set X ⊆ [0, 1] (e.g., Brendle [11] or Miller [28]).By a result of Galvin and Miller (1984) [16], if Y ⊆ X is not Fσ or notGδ, then (X \ Y ) ∪ (Y + 1) 6∈ S1(Ω, Γ). This solves both problems atonce.

The striking observation we get from this journey of topological ap-proximations to the Minimal Tower Problem is that, despite the factthat the purely combinatorial problem is very difficult, if we add to ittopological structure we get a negative answer quite easily.

It is surprising, though, that the following related problem remainsopen.

Problem 5.4. Is(

ΩΓ

)

=(

ΩT

)

?

The critical cardinality of(

ΩT

)

is p [42], so both properties have thesame critical cardinality.

Remark 5.5. The reader interested in another topological study whichwas inspired by (and is related to) the Minimal Tower Problem is re-ferred to Machura’s [24].

5.6. Known implications and critical cardinalities. Having thisnew notion of rich covers, namely τ -covers, it is interesting to try andadd it to Scheepers’ framework of selection principles. This yields manymore properties, but again some of them can be proved to be equivalent[48].

As already mentioned, a basic tool to prove nonimplications is thecomputation of critical cardinalities: If P and Q are properties withnon(P ) < non(Q) consistent, then Q does not imply P . The criticalcardinalities of most of the new properties were found in [48] and inShelah, et. al. [42]. Still, 6 critical cardinalities remained unsettled.These remaining cardinalities were addressed by Mildenberger, Shelah,et. al., [27]. We give here one example of that treatment, and quotethe main results. (Everything in the remainder of this subsection isquoted, without further notice, from [27]).

Let CΓ, CT, and CΩ denote the collections of clopen γ-covers, τ -covers, and ω-covers of X, respectively. Recall that, since we are deal-ing with sets of reals, we may assume that all open covers are countable.Restricting attention to countable covers, we have the following, wherean arrow denotes inclusion:

BΓ → BT → BΩ → B↑ ↑ ↑ ↑Γ → T → Ω → O↑ ↑ ↑ ↑

CΓ → CT → CΩ → C


As each of the properties Π(· , ·), Π ∈ S1, Sfin, Ufin, is monotonic inits first variable, we have that for each x, y ∈ Γ, T, Ω,O,

Π(Bx,By) → Π(x, y) → Π(Cx, Cy)

(here CO := C and BO := B). Consequently,

non(Π(Bx,By)) ≤ non(Π(x, y)) ≤ non(Π(Cx, Cy)).

Definition 5.6. We use the short notation ∀∞ for “for all but finitelymany” and ∃∞ for “there exist infinitely many”.

(1) A ∈ 0, 1N×N is a γ-array if (∀n)(∀∞m) A(n, m) = 1.

(2) A ⊆ 0, 1N×N is a γ-family if each A ∈ A is a γ-array.

(3) A family A ⊆ 0, 1N×N is finitely τ -diagonalizable if there existfinite (possibly empty) subsets Fn ⊆ N, n ∈ N, such that:(a) For each A ∈ A: (∃∞n)(∃m ∈ Fn) A(n, m) = 1;(b) For each A, B ∈ A:

Either (∀∞n)(∀m ∈ Fn) A(n, m) ≤ B(n, m),or (∀∞n)(∀m ∈ Fn) B(n, m) ≤ A(n, m).

Using the dictionary of Section 5.3, one can prove the following.(Notice that 0, 1N×N is topologically the same as the Cantor space0, 1N.)

Theorem 5.7. For a set of reals X, the following are equivalent:

(1) X satisfies Sfin(BΓ,BT); and

(2) For each Borel function Ψ : X → 0, 1N×N, if Ψ[X] is a γ-family, then it is finitely τ -diagonalizable.

The corresponding assertion for Sfin(CΓ, CT) holds when “Borel” isreplaced by “continuous”.

Lemma 5.8. The minimal cardinality of a γ-family which is not finitelyτ -diagonalizable is b.

(To appreciate the result in Lemma 5.8, the reader may wish to tryproving it for a while.) Having Theorem 5.7 and Lemma 5.8, we getthat

b = non(Sfin(BΓ,BT)) ≤ non(Sfin(Γ, T)) ≤ non(Sfin(CΓ, CT)) = b,

and therefore non(Sfin(Γ, T)) = b.Notice that Theorem 5.7 reduced the original topological question

into a purely combinatorial question. Its solution in Lemma 5.8 maybe of independent interest to those working on this sort of pure com-binatorics.

It follows that non(S1(Γ, T)) = b, and using a similar approach, itcan be proved that non(S1(T, T)) = t, and non(Sfin(T, T)) = mins, b.

20 BOAZ TSABAN

It is not difficult to see that non(Sfin(T, Ω)) = non(Sfin(T,O)), callthis joint cardinal o, the o-diagonalization number ; the reason for thisto be explained soon. The surviving properties (in the open case)appear in Figure 6, with their critical cardinalities, and serial numbers(for later reference). The new cardinalities computed in [27] are framed.

Ufin(Γ, Γ)

b (18)// Ufin(Γ, T)

maxb, s (19)// Ufin(Γ, Ω)

d (20)// Ufin(Γ,O)

d (21)

Sfin(Γ, T)

b (12)//

77nnnnn

Sfin(Γ, Ω)

d (13)

88rrrr

S1(Γ, Γ)b (0)

88rr

rr

rrr

rrr

rr

rrr

rrr

r

//S1(Γ, T)

b (1)

99rrrrr

// S1(Γ, Ω)d (2)

88r

rrrr

// S1(Γ,O)d (3)

::u

uu

uu

uu

uu

uu

uu

uu

u

Sfin(T, T)

mins, b (14)//

OO

Sfin(T, Ω)

d (15)

OO

S1(T, Γ)t (4)

//

OO

S1(T, T)

t (5)

OO

99ssss

//S1(T, Ω)

?(o) (6)

OO

99sssss

//S1(T,O)

?(o) (7)

OO

Sfin(Ω, T)

p (16)

OO

// Sfin(Ω, Ω)

d (17)

OO

S1(Ω, Γ)p (8)

OO

// S1(Ω, T)p (9)

OO

77pppp

// S1(Ω, Ω)cov(M) (10)

OO

77ooo

// S1(O,O)cov(M) (11)

OO

Figure 6. The Scheepers diagram, enhanced with τ -covers

By Figure 6,

cov(M) = non(S1(O,O)) ≤ non(S1(T,O)) ≤ non(S1(Γ,O)) = d,

thus cov(M) ≤ o ≤ d.

Definition 5.9. A τ -family A is o-diagonalizable if there exists a func-tion g : N → N, such that:

(∀A ∈ A)(∃n) A(n, g(n)) = 1.

As in Theorem 5.7, S1(BT,B) and S1(CT, C) have a natural combi-natorial characterization. This characterization implies that o is equalto the minimal cardinality of a τ -family that is not o-diagonalizable. Adetailed study of o is initiated in [27]; the main results being that con-sistently o < minh, s, b, and in many standard models of set theory(Cohen, Random, Hechler, Laver, Mathias, and Miller), cov(M) = o.

Problem 5.10 (Mildenberger, Shelah, et. al. [27, Problem 4.13]). Isit consistent (relative to ZFC) that cov(M) < o?


It is worthwhile mentioning that this problem, which originated fromthe topological studies of the minimal tower problem, is of similar fla-vor: It is well-known that if p = ℵ1, then t = ℵ1 too. We have a similarassertion for cov(M) and o: If cov(M) = ℵ1 < b, then cov(M) = o.

5.7. A table of open problems. It is possible that the diagram inFigure 6 is incomplete: There are many unsettled possible implicationsin it. After [48, 42], there remained 76 (!) potential implications whichwere not proved or ruled out. The study made recently in [27] anddescribed in the previous section ruled out 21 of these implications, sothat 55 implications remain unsettled. The situation is summarized inTable 1, which updates the corresponding table given in [50].

Each entry (i, j) (ith row, jth column) contains a symbol. Xmeansthat property (i) in Figure 6 implies property (j) in Figure 6. × meansthat property (i) does not (provably) imply property (j), and ? meansthat the corresponding implication is still unsettled.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0 X X X X × × × × × × × × X X × ? × × X X X X

1 ? X X X × × × × × × × × X X × ? × × ? X X X

2 × × X X × × × × × × × × × X × ? × × × × X X

3 × × × X × × × × × × × × × × × × × × × × × X

4 X X X X X X X X × × ? ? X X X X × ? X X X X

5 ? X X X ? X X X × × ? ? X X X X × ? ? X X X

6 × × X X × × X X × × ? ? × X × X × ? × × X X

7 × × × X × × × X × × × ? × × × × × × × × × X

8 X X X X X X X X X X X X X X X X X X X X X X

9 ? X X X ? X X X ? X X X X X X X X X ? X X X

10 × × X X × × X X × × X X × X × X × X × × X X

11 × × × X × × × X × × × X × × × × × × × × × X

12 ? ? ? ? × × × × × × × × X X × ? × × ? X X X

13 × × × × × × × × × × × × × X × ? × × × × X X

14 ? ? ? ? × × × × × × × × X X X X × ? ? X X X

15 × × × × × × × × × × × × × X × X × ? × × X X

16 ? ? ? ? ? ? ? ? ? ? ? ? X X X X X X ? X X X

17 × × × × × × × × × × × × × X × X × X × × X X

18 × × × × × × × × × × × × × ? × ? × × X X X X

19 × × × × × × × × × × × × × ? × ? × × × X X X

20 × × × × × × × × × × × × × ? × ? × × × × X X

21 × × × × × × × × × × × × × × × × × × × × × X

Table 1. Known implications and nonimplications

Problem 5.11. Settle any of the remaining 55 implication in Table 1.

5.8. The Minimal Tower Problem revisited. The study of τ -coverswas motivated by a combinatorial problem. Interestingly, it rewardedus with a closely related purely combinatorial problem. One of thedifficulties with treating τ -covers is the following: Unlike the case of

22 BOAZ TSABAN

ω-covers or γ-covers, it could be that U is a τ -cover of X which refinesanother cover V of X, but V is not a τ -cover of X. This led to theintroduction of the following close relative [48].

A family Y ⊆ [N]ℵ0 is linearly refinable if for each y ∈ Y there exists

an infinite subset y ⊆ y such that the family Y = y : y ∈ Y islinearly (quasi)ordered by ⊆∗. A cover U of X is a τ ∗-cover of X if itis large, and h[X] (where h is the Marczewski characteristic functionof U , see Section 5.3) is linearly refinable. Let T∗ denote the collectionof all countable open τ ∗-covers of X.

If we restrict attention to countable covers (this does not make adifference for sets of reals), then T ⊆ T∗ ⊆ Ω. Let p∗ = non(

(

ΩT∗

)

).Then by the dictionary (Section 5.3),

p∗ = min|Y | : Y ⊆ [N]ℵ0 is centered but not linearly refineable,

and we have the following interesting theorem (exercise):

Theorem 5.12. p = minp∗, t.

Thus p < t implies the somewhat more pathological situation p∗ < t.If p∗ is not provably equal to p then Theorem 5.12 may be regarded asa partial (but easy) solution to the Minimal Tower Problem.

Problem 5.13. Is p = p∗?

As we have mentioned before, the closely related problem whetherp = non(

(

ΩT

)

) has a positive answer in [42].

6. Some connections with other fields

6.1. Ramsey theory. Recall that Ramsey’s Theorem, often writtenas

(∀n, k) ℵ0 → (ℵ0)n

k ,

asserts that for each n, k, and a countably infinite set I: For eachcoloring f : [I]n → 1, . . . , k, there exists a color j and an infiniteJ ⊆ I such that f [J ]n ≡ j.

This motivates the following prototype for Ramsey theoretic hy-potheses [36].

A → (B)nk : For each U ∈ A and f : [U ]n → 1, . . . , k, there exists j and

V ⊆ U such that V ∈ B and f [V]n ≡ j.

Using this notation, Ramsey’s Theorem is (∀n, k) [N]ℵ0 → ([N]ℵ0)nk .

This prototype can be applied to the case where A and B are collectionsof rich covers, in accordance to the major theme of Ramsey theory thatoften, when rich enough structures are split into finitely many pieces,one of the pieces contains a rich substructure (Furstenberg).


The simplest case of the mentioned Ramseyan property is wheren = 1 (so that we color the elements of the given member of A ratherthan finite subsets of it). This case is well understood: Fix a spaceX. Since a finite partition of an infinite set must contain an infiniteelement, and since every infinite subset of a γ-cover of X is again aγ-cover of X, we have that

(∀k) Γ → (Γ)1k.

It is less trivial but still not difficult to show that

(∀k) Ω → (Ω)1k

also holds, and since each τ -cover is an ω-cover (which in turn is a largecover), and each large subcover of a τ -cover is again a τ -cover, we havethat

(∀k) T → (T)1k.

The corresponding assertions for Borel or clopen (instead of open) cov-ers also hold for the same reasons, and it can be shown that nothingmore can be said (except for what trivially follows from the above as-sertions), concerning the partition property A → (B)1

k where A,B ∈O, Ω, T, Γ. We now turn to the case that n > 1.

Ramsey’s Theorem is exactly the same as

(∀n, k) Γ → (Γ)n

k .

However, if we substitute other classes of covers for A and B the prop-erty may not hold for all X. An elegant example is the following.

Theorem 6.1 (Just-Miller-Scheepers-Szeptycki [36, 20]). The follow-ing properties are equivalent:

(1) S1(Ω, Ω),(2) Ω → (Ω)2

2; and(3) (∀n, k) Ω → (Ω)n

k .

In other words, X satisfies S1(Ω, Ω) if, and only if, whenever we colorwith 2 colors the edges of a complete graph whose vertices are elementsof an ω-cover of X, we can find a complete monochromatic subgraphwhose vertices form an ω-cover of X.Corollary 6.2.

(1) S1(Ω, Γ) is equivalent to Ω → (Γ)22.

(2) S1(Ω, T) implies Ω → (T)22.

And in both cases we can use any n and k as a superscript and asubscript.

24 BOAZ TSABAN

Proof. (⇒) for (1) and (2): Assume that Ω ⊇ A and X satisfiesS1(Ω,A). Then X satisfies S1(Ω, Ω) as well as

(

ΩA

)

. By Theorem 6.1, X

satisfies Ω → (Ω)22, but since X satisfies

(

ΩA

)

and a (complete) subgraphof a monochromatic graph is monochromatic, X satisfies Ω → (A)2

2.(⇐) for (1): Clearly, Ω → (Γ)2

2 implies(

ΩΓ

)

, which we know is thesame as S1(Ω, Γ).

Problem 6.3. Is S1(Ω, T) equivalent to Ω → (T)22 ?

Even for n = k = 2, Ramsey’s Theorem cannot be extended to ℵ1.To see this, consider a set of reals X of size ℵ1, and a wellordering ≺ ofX. Give a pair x, y ⊆ X the color 1 if < and ≺ agree on (x, y), and0 otherwise. Then a monochromatic subgraph would give rise to an<-chain of reals of size ℵ1, which is impossible, since between each twoelements of the chain there is a rational number. (This nice argumentis due to Sierpinski.)

There are some other ways to extend Ramsey’s Theorem to largercardinals, sometimes taking a quick route to large cardinals, see [18]for a survey. But Theorem 6.1 and its relatives suggest other general-izations. If we forget about the topology and consider arbitrary (butcountable) covers of the given space (that is, set), then Theorem 6.1implies the following. For an infinite cardinal κ, let Ωκ denote thecollection of all countable ω-covers of κ.

Theorem 6.4 (Scheepers [37]). If κ < cov(M), then (∀n, k) Ωκ →(Ωκ)n

k , where Ωκ is the collection of countable ω-covers of κ. (Moreover,for each n and k the bound cov(M) is tight.)

This is so because non(S1(Ω, Ω)) = cov(M). This is a nice andtypical case where the results in the topological studies, which weremotivated by results from pure infinite combinatorics, often project tonew results in pure infinite combinatorics.

6.2. Countably distinct representatives and splittability. Themain tools for proving Ramsey theoretic results of the flavor of theprevious section are the properties of the following form:

CDR(A,B): For each sequence Unn∈N of elements of A, there exist count-able, pairwise disjoint elements Vn ⊆ Un, n ∈ N, such thatVn ∈ B for all n.

Split(A,B): For each U ∈ A, there exist disjoint V1,V2 ⊆ U such thatV1,V2 ∈ B.

Clearly, CDR(A,B) implies Split(A,B), of which an almost completeclassification was carried in [47], when A,B ∈ Ω, T, Γ, Λ. As before,


some of the properties are trivial, and several equivalences are provableamong the remaining properties.

The following dictionary is the key behind the classification, wherethe negation of each property corresponds to some combinatorial prop-erty of h[X] where h is the Marczewski characteristic function (Section5.3) of U for a cover U witnessing the failure of that property.

Property h[X]

¬Split(Λ, Λ) reaping family¬Split(Ω, Λ) ultrafilter base¬Split(Ω, Ω) ultrafilter subbase¬Split(T, T) simple P -point base

The results of the classification are summarized in the following the-orem.

Theorem 6.5 ([47]). No additional implication (except perhaps thedotted ones) is provable.

Split(Λ, Λ) // Split(Ω, Λ) // Split(T, T)

Split(Ω, Ω)

OO

Split(Ω, T)

OO

(1)

''

(2)

(3)

__

Split(Ω, Γ)

OO

77ooooooooooo

// Split(T, Γ)

OO

Problem 6.6. Is the dotted implication (1) (and therefore (2) and (3))in the diagram true? If not, then is the dotted implication (3) true?

6.3. An additivity problem. With regards to the additivity (preser-vation under taking finite unions) and σ-additivity (countable unions),the following is known (Xmeans that, in the figure of Theorem 6.5,the property in this position is σ-additive, and × means that it is notadditive).

? // X // X

×

OO

×

OO

×

OO

99s

ss

ss

s// X

OO

26 BOAZ TSABAN

Thus, the only unsettled problem is the following:

Problem 6.7. Is Split(Λ, Λ) additive?

In Proposition 1.1 of [47] it is shown that for a set of reals X (in fact,for any hereditarily Lindelof space X), each large open cover of X con-tains a countable large open cover of X. Consequently, using standardarguments [47], the problem is closely related to the the following one(where [N]ℵ0 is the space of all infinite sets of natural numbers, withthe topology inherited from P (N), the latter identified with 0, 1N).

Problem 6.8. If R denotes the sets of reals X such that each contin-uous image of X in [N]ℵ0 is not reaping, then is R additive?

Zdomsky has proved the following surprising result concerning theHurewicz and Menger properties.

Theorem 6.9 (Zdomsky [55]). Assume that u < g. Then Ufin(O, Γ) =Split(Λ, Λ), and Ufin(O, Ω) = Ufin(O,O).

Since Ufin(O, Γ) is easily seen to be countably additive, it followsthat the answer to Problem 6.7 is consistently positive.

6.4. Arkhangel’skii duality theory. Consider C(X), the space ofcontinuous real-valued functions f : X → R, as a subspace of RX (theTychonoff product of X many copies of R). This is often called thetopology of pointwise convergence and sometimes denoted Cp(X) ratherthan C(X).

The object C(X) is very complicated from a topological point ofview, in fact, even the behavior of the closure operator in this space iscomplicated. To this end, a comprehensive duality theory was devel-oped by Arkhangel’skii and his followers, which translates propertiesof C(X) into properties of X, which are easier to work with.

For example, recall that a space Y is Frechet-Urysohn if for eachA ⊆ Y and x ∈ A, there is a sequence ann∈N in A such thatlimn→∞ an = x. In their celebrated 1982 paper [17], Gerlits and Nagyproved that C(X) is Frechet-Urysohn if, and only if, X satisfies

(

ΩΓ

)

(or, equivalently, S1(Ω, Γ)).An interesting example of the applicability of infinite-combinatorial

methods for these questions is the following. In 1992, at a seminar inMoscow, Reznicenko introduced the following property: A space Y isweakly Frechet-Urysohn if, for each A ⊆ Y and x ∈ A \ A, there existfinite disjoint sets Fn ⊆ A, n ∈ N, such that for each neighborhood Uof x, Fn ∩ U 6= ∅ for all but finitely many n.

In [22], Kocinac and Scheepers made the following conjecture, whichis now a theorem.


Theorem 6.10 ([46]). The minimal cardinality of a set of reals X suchthat C(X) does not have the weak Frechet-Urysohn property is b.

The surprising thing is that its proof only required a translation intothe language of combinatorics and an application of an existing result:It was known that this minimal cardinality is at least b. A result ofSakai [35] can be used to prove that if C(X) is weakly Frechet-Urysohn,then a continuous image of X cannot be a subbase for a non-feeble filteron N (see [9] for the definition of non-feeble filter). Now it remained toapply a result of Petr Simon which tells that there exists a non-feeblefilter with base of size b.

7. Conclusions

The theory emerging from the systematic study of diagonalizationsof covers in a unified framework is not only aesthetically pleasing, butis also useful in turning otherwise ingenious ad hoc arguments intonatural explanations. For this a good terminology is required, andthe major part of this was already suggested by Scheepers’ selectionprototypes.

This study has connections and applications in several related fields,like Ramsey theory, function spaces, and topological groups. The usageof infinite-combinatorial methods in this theory has proved successful,and is often the “correct” tool to investigate these problems. Thisapproach sometimes rewords by implying interesting results in the fieldof pure infinite-combinatorics.

While with regards to some of the investigations concerning the clas-sical types of covers the picture is rather complete now, there remainsmuch to be explored with regards to the new types of covers and theirconnection to the related fields.

We have only given a tiny sample of each theme. The reader isreferred to [40, 21, 50] to have a more complete picture of the frameworkand the open problems it poses.

7.1. Acknowledgments. We thank Tomasz Weiss for the proof ofProposition 2.2, and Lubomyr Zdomsky for reading the paper and mak-ing several interesting comments.

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Einstein Institute of Mathematics, The Hebrew University of Jerusalem,

Givat Ram, 91904 Jerusalem, Israel

E-mail address : [email protected]

URL: http://www.cs.biu.ac.il/~tsaban

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