We examine the class of spaces in which the second player has a winning
strategy in the open--open game. We show that this spaces are not universally
Kuratowski-Ulam. We also show that the games G and G7 introduced by P. Daniels,
K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205--220] are not equivalent

Bartoszyński

Daniels

Fremlin

Just

Kucharski

Kuratowski

Pawlikowski

Scheepers

Tsaban

English

arXiv

We examine the class of spaces in which the second player has a winning strategy in the open-open game. We show that this spaces are not universally Kuratowski–Ulam. We also show that the games G and G7 introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205–220] are not equivalent

Kalemba, Piotr

Kucharski, Andrzej

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Universally Kuratowski-Ulam spaces and open-open games  Author: Piotr Kalemba, Andrzej Kucharski  Citation style: Kalemba Piotr, Kucharski Andrzej. (2015). Universally Kuratowski-Ulam spaces and open-open games. "Annales Mathematicae Silesianae" (Nr 29 (2015), s. 85-92), doi 10.1515/amsil-2015-0007  Annales Mathematicae Silesianae 29 (2015), 85–92Prace Naukowe Uniwersytetu Śląskiego nr 3332, KatowiceDOI: 10.1515/amsil-2015-0007UNIVERSALLY KURATOWSKI–ULAM SPACESAND OPEN-OPEN GAMESPiotr Kalemba, Andrzej KucharskiAbstract. We examine the class of spaces in which the second player has awinning strategy in the open-open game. We show that this spaces are not uni-versally Kuratowski–Ulam. We also show that the games G and G7 introducedby P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205–220]are not equivalent.1. IntroductionFirst we shall recall some game introduced in [2] called G2. Let X bea topological space equipped with a topology T and let B ⊆ T be its base.The length of the game is ω. Two players I and II take turns playing. At then-th move II chooses a family Pn consisting of open non-empty subset of Xsuch that cl⋃Pn = X, then I picks a Vn ∈ Pn. I wins iff cl⋃n∈ω Vn = X.Otherwise player II wins. Denote by Dcov a collection of families F consistingof open sets with cl⋃F = X. We say that σcov : (⋃Dcov)<ω → Dcov is awinning strategy for player II in the game G2 whenever, for any sequenceU0, U1, . . . consisting of non-empty open subsets with U0 ∈ σcov(∅) = P0 ∈Dcov and Un ∈ σcov(U0, U1, . . . , Un−1) = Pn ∈ Dcov, for all n ∈ ω, there holdscl⋃n∈ω Un 6= X.Received: 10.03.2015. Revised: 6.06.2015.(2010) Mathematics Subject Classification: 54G20, 91A44, 54F99.Key words and phrases: II-favorable space, uK-U space, tiny sequence.Andrzej Kucharski thanks European Science Foundation for their support through thegrant 3007 within the INFTY program.86 Piotr Kalemba, Andrzej KucharskiIn the paper [2] the authors introduced an open-open game. We say thatG is an open-open game of length ω if two players take turns playing; a roundconsists of player I choosing a non-empty open set U ⊆ X and player IIchoosing a non-empty open V ⊆ U ; I wins if the union of II’s open sets isdense in X, otherwise II wins. Suppose that there exists a functionsop :⋃{T n : n ≥ 0} → Tsuch that for each sequence V0, V1, . . . consisting of non-empty elements ofT with sop(V0) ⊆ V0 and sop(V0, V1, . . . , Vn) ⊆ Vn, for all n ∈ ω, there holdscl⋃n∈ω Vn 6= X. Then the function sop is called a winning strategy for IIplayer in the open-open game and we say that the space X is II-favorable.It is known [2] that the open-open game G is equivalent to G2. We consideronly games with the length equal to ω. In [2] the authors introduced a gameG7 which is played as follows: In the n-th inning II chooses On, a family ofopen sets with⋃On dense in X. I responds with Tn, a finite subfamily of On;I wins if⋃n∈ω Tn is dense subset of X; otherwise, II wins.According to A. Szymański [13] a sequence {Pn : n ∈ ω} of open familiesin X is a tiny sequence if(1)⋃Pn is dense in X for all n ∈ ω(2) if Fn is a finite subfamily of Pn for each n ∈ ω then⋃{⋃Fn : n ∈ ω} isnot dense in X.We call a sequence {Pn : n ∈ ω} of open families in X a 1-tiny sequence if(1)⋃Pn is dense in X for all n ∈ ω(2) if Fn is a member of Pn for each n ∈ ω then⋃{Fn : n ∈ ω} is not densein X.M. Scheepers used in the paper [12] negation of the existence of tiny se-quence, and 1-tiny sequence - called these properties Sfin(D,D) and S1(D,D)respectively. In this paper we refer to notions tiny sequence and 1-tiny se-quence, because in some situations (Theorem 1.1 and 1.2) we can define them.Recall another game G4 introduced in [2]. In the n-th inning player Ichooses finite open family An. Player II responds with a finite, open familyBn with |Bn| = |An| and for each V ∈ An there exists W ∈ Bn such thatW ⊆ V . I wins if ⋃n∈ω⋃Bn is dense subset of X; otherwise, II wins. Onecan prove that the game G7 is equivalent to the game G4 in a way similar tothe proof of the equivalence between games G and G2.From now on we consider only c.c.c. spaces.Theorem 1.1 (M. Scheepers [12, Theorem 2]). II has a winning strategyin the game G7 if and only if there exists a tiny sequence.Universally Kuratowski–Ulam spaces and open-open games 87Theorem 1.2 (M. Scheepers [12, Theorem 14]). Player II has a winningstrategy in the game G2 if and only if there exists a 1-tiny sequence.2. The main resultsRecall that X is called a II-favorable space if player II a has winningstrategy in the game G. If player I has a winning strategy in the game G thenwe say that the space is I-favorable.The following theorem was proven by K. Kuratowski and S. Ulam, see [9].In order to formulate it, let us recall that: a pi-base is a family of open,nonempty sets such that any open set contains a set from this family, andthe pi-weight of a space is the smallest cardinality of a pi-base in this space.Let X and Y be topological spaces such that Y has countable pi-weight. IfE ⊆ X × Y is a nowhere dense set, then there is a meager set P ⊆ X suchthat the section Ex = {y : (x, y) ∈ E} is nowhere dense in Y for each pointx ∈ X \ P .A space Y is universally Kuratowski–Ulam (for short, uK-U space), when-ever for a topological space X and a nowhere dense set E ⊆ X × Y the set{x ∈ X : {y ∈ Y : (x, y) ∈ E} is not nowhere dense in Y }is meager in X, see D. Fremlin [6] (compare [3]). In the paper [7] authorshave shown that a compact I-favorable space is universally Kuratowski–Ulamand posed a question: Does there exist a compact universally Kuratowski–Ulam space which is not I-favorable? We will partially answer to this question,namely we will prove that a II-favorable space is not universally Kuratowski–Ulam space.Theorem 2.1. Let X be a dense in itself space with a pi-base B = ⋃n∈ω Bn,where Bn is a maximal family of pairwise disjoint open sets for n ∈ ω and letY be II-favorable c.c.c. space. Then the Kuratowski–Ulam theorem does nothold in X × Y .Proof. By Theorem 1.2 there is a 1-tiny sequence {Pn : n ∈ ω}. Since thespace Y satisfies c.c.c., we can assume that each Pn+1 is a countable, open,pairwise disjoint family. We can also assume that every Pn+1 is a refinementof Pn, i.e. each member of Pn+1 is a subset of a member of Pn. Let {V nσ :σ ∈ nN} be an enumeration of the family Pn such that for each τ ∈ n−1N,{V nτ_k : k ∈ N} = Pn.88 Piotr Kalemba, Andrzej KucharskiWe can assume that Bn+1 is a refinement of Bn and |{V ∈ Bn+1 : V ⊆U}| ≥ ω for each U ∈ Bn. For each n ∈ N fix a function fn : Bn → nN suchthat for a fixed U ∈ Bn we have(2.1) {fn+1(V ) : V ∈ Bn+1 and V ⊆ U} = fn(U)_N.Therefore, there holds the condition:(2.2) if V ⊂ U then fn+1(V ) ⊃ fn(U) for every V ∈ Bn+1 and U ∈ Bn.Consider an open setF =⋃{⋃{U × V nfn(U) : U ∈ Bn} : n ∈ N}.We shall show that F is dense and Fx = {y ∈ Y : (x, y) ∈ F} is not densefor each x ∈ X. If x ∈ X \ ⋂{⋃Bn : n ∈ N} then it is easy to see thatEx is not dense. If x ∈⋂{⋃Bn : n ∈ N} then by condition (2.2) there isσ ∈ NN such that for each n ∈ N there exists Un ∈ Bn with fn(Un) = σ|n andx ∈ ⋂{Un : n ∈ N}, hence Fx = ⋃{V nσ|n : n ∈ N}. Since V nσ|n ∈ Pn for eachn ∈ N and {Pn : n ∈ ω} is a 1-tiny sequence the set⋃{V nσ|n : n ∈ N} is notdense.Now we show that F is a dense set. Let U×W be any open set. Since B is api-base there are n ∈ N and U0 ∈ Bn such that U0 ⊆ U . Let σ = fn(U0), since{V n+1σ_k : k ∈ N} is a dense family, we get that W ∩ V n+1σ_k 6= ∅ for some k ∈ N.By (1), we may take U1 ⊆ U0 such that U1 ∈ Bn+1 and fn+1(U1) = σ_k.Thus U1 × V n+1fn+1(U1) ∩ U ×W 6= ∅. Since R with natural topology satisfies assumption of the above theoremand every universally Kuratowski–Ulam space is c.c.c. space, we get the fol-lowing theorem.Theorem 2.2. A II-favorable space is not universally Kuratowski–Ulamspace.Following [10, pp. 86–91] recall category measure space. If X is a topo-logical space with finite measure µ defined on the σ-algebra S of sets havingthe Baire property, and if µ(E) = 0 if and only if E is of a meager set,then (X,S, µ) is called a category measure space. An example of a regularBaire space which is a category measure space, is an open interval (0, 1) withLebesgue measure µl and density topology Td, see [10]. For density topologyand measurable set A ⊆ (0, 1) the following conditions are equivalent:Universally Kuratowski–Ulam spaces and open-open games 89(1) µl(A) = 0,(2) A is closed and nowhere dense.In the space ((0, 1), Td) there is a 1-tiny sequence but there is no tiny sequence.Indeed, define a 1-tiny sequence in the following way: let Pn = {U : U ∈Td and µl(U) ≤ 13n }. If {Un : n ∈ N} is a family chosen by player I thenµl(⋃{Un : n ∈ N}) ≤ 12 . Therefore {Un : n ∈ N} is not a dense family. Nowassume that there exists a tiny sequence {Pn : n ∈ N}. In each stage we choosea finite subfamily Rn ⊂ Pn such that µl(⋃{⋃Ri : i ≤ n}) ≥ 1− 1n , hence weget a dense family⋃{Rn : n ∈ N}.The authors of the paper [2] posed a question (Question 4.3): Does aplayer have a winning strategy in the game G if and only if the same playerhas a winning strategy in the game G7. The author of paper [12] showed thatif cov(M) < d the answer is NO. We show that games G and G7 are notequivalent.Corollary 2.3. The game G is not equivalent to the game G7.Proof. By Theorem 1.2 a winning strategy of II player in the game Gis equivalent to the existence of a 1-tiny sequence and by Theorem 1.1 theexistence of a winning strategy of player II in the game G7 is equivalent tothe existence of a tiny sequence. Since in the space ((0, 1), Td) there is a 1-tinysequence but there is no tiny sequences we get that games G and G7 are notequivalent. Since the game G7 is equivalent to the game G4, we get the following:Corollary 2.4. The game G is not equivalent to the game G4.3. Some remarksIt is known that on the ω1 with discrete topology II player has a winningstrategy in the game G7, but one can pose a question:Is it possible to construct a tiny sequence {Pn : n ∈ ω} on a discrete spaceof the size ω1 with |Pn| = ω for all n ∈ ω ?The following Remark 3.1 gives us the answer - it is possible if and only ifthe dominating number is equal ω1. This is reformulation of well know resultsabout critical cardinal number, see W. Just, A.W. Miller, M. Scheepers andP.J. Szeptycki [5]; D. Fremlin, A.W. Miller [4] and B. Tsaban [14].Recall that f ≤∗ g denotes that for almost all n ∈ ω holds f(n) ≤ g(n),where f, g are functions defined on natural numbers. A family R ⊆ ωω is90 Piotr Kalemba, Andrzej Kucharskia dominating family if for each f ∈ ωω there is g ∈ R such that f ≤∗ g. Thedominating number d is the smallest cardinality of a dominating family:d = min{|R| : R is dominating }.Remark 3.1. The smallest cardinality κ such that there exists a tinysequence {Pn : n ∈ ω} on the discrete space of the size κ with |Pn| = ω for alln ∈ ω is equal to d.Proof. Let X be any discrete space for which there exists a tiny sequence{Pn : n ∈ ω}. We can assume that every Pn is a partition of X into countablymany blocks {Xn0 , Xn1 , . . .}, so we may define for each x ∈ X a functionfx : ω → ω in the following way: fx(n) = k whenever x ∈ Xnk . Take anarbitrary function f : ω → ω, and any x ∈ X \⋃{⋃{Xnk : k ≤ f(n)} : n < ω},then f is dominated by the function fx. It shows that {fx : x ∈ X} is adominating family, hence |X| ≥ d.Now, let F ⊂ ωω be a dominating family of the cardinality d. Withoutloss of generality assume that for each function f : ω → ω there is g ∈ F suchthat f(n) < g(n) for all n < ω. We treat F as a discrete topological space.For n, k ∈ ω put Ank = {f ∈ F : f(n) ≤ k} and set Pn = {Ank : k < ω}. Ofcourse, each family Pn is increasing and has the union equal to F . From eachPn take some single Anf(n) where f : ω → ω. If⋃{Anf(n) : n < ω} was equal toF , then it would contain such a function g that g(n) > f(n) for all n ∈ ω,but it is not the case. Therefore {Pn : n ∈ ω} is a tiny sequence. Recall a definition of a Baire number cov(M) for the idealM of meagersubsets of real line R:cov(M) = min{|A| : A ⊆M and⋃A = R}.T. Bartoszyński [1] proved that cov(M) is the cardinality of the smallestfamily F ⊆ ωω such that∀(g ∈ ωω) ∃(f ∈ F) ∀(n ∈ ω) f(n) 6= g(n).We get another well known characterization of such families by a 1-tiny se-quence.Remark 3.2. The smallest cardinality κ such that there exists a 1-tinysequence {Pn : n ∈ ω} on the discrete space of the size κ with |Pn| = ω for alln ∈ ω is equal to cov(M).Universally Kuratowski–Ulam spaces and open-open games 91We give the proof for the sake of completeness. We shall prove that thesmallest cardinality of a family F ⊆ ωω such that(∗) ∀(g ∈ ωω) ∃(f ∈ F) ∀(n ∈ ω) f(n) 6= g(n)is equal to the smallest cardinality κ such that there exists a 1-tiny sequence{Pn : n ∈ ω} on the discrete space κ with |Pn| = ω for all n ∈ ω.Proof. Let F = {fα : α < κ} ⊆ ωω be a family with the property (∗).Define Ain = {f ∈ F : f(i) = n} for every i, n ∈ ω. Let Pi = {Ain : n ∈ ω} fori ∈ ω. We will show that {Pi : i ∈ ω} is a 1-tiny sequence. Assume that wehave chosen Aini ∈ Pi for each i ∈ ω. Define a function g(i) = ni for i ∈ ω.Since F satisfies (∗) there is f ∈ F such that f(i) 6= g(i) for each i ∈ ω.Therefore we get f ∈ F \⋃{Aini : i ∈ ω}.Let {Pn : n ∈ ω} be a 1-tiny sequence with |Pn| = ω and⋃Pn = κ foreach n ∈ ω. We can assume that each Pn consists of pairwise disjoint subsetsof κ. Let {Ank : k ∈ ω} be a enumeration of Pn. We define a function fx ∈ ωωfor each x ∈ κ in the following way: fx(i) = n, where x ∈ Ain for each i ∈ ω.The family {fx : x ∈ κ} satisfies (∗). Indeed, let g ∈ ωω be any function. Since{Pn : n ∈ ω} is a 1-tiny sequence, choose x ∈ κ \⋃{Aig(i) : i ∈ ω}. Finally,observe that fx(i) 6= g(i) for every i ∈ ω. We shall recall definition of the bounding numberb = min {|F| : F ⊆ ωω and ∀(g ∈ ωω) ∃(f ∈ F) ¬(f ≤∗ g))} .We say that a sequence {Pn : n ∈ ω} of open families in X is a b-tinysequence if(1)⋃Pn is dense in X for all n ∈ ω;(2) if Fn is a finite subfamily of Pn for each n ∈ ω, then there exists strictlyincreasing sequence {ni : i ∈ ω} such that⋃{⋃Fni : i ∈ ω}is not dense in X.We get the next reformulation of the bounding number.Remark 3.3. The smallest cardinality κ such that there exists a b-tinysequence {Pn : n ∈ ω} on the discrete space of the size κ with |Pn| = ω forall n ∈ ω is equal to b.92 Piotr Kalemba, Andrzej KucharskiAcknowledgement. The authors are indebted to the Referee for verycareful reading of the paper and valuable comments.References[1] Bartoszyński T., Combinatorial aspects of measure and category, Fund. Math. 127(1987), no. 3, 225–239.[2] Daniels P., Kunen K., Zhou H., On the open-open game, Fund. Math. 145 (1994), no.3, 205–220.[3] Fremlin D., Natkaniec T., Recław I., Universally Kuratowski–Ulam spaces, Fund.Math. 165 (2000), no. 3, 239–247.[4] Fremlin D., Miller A.W., On some properties of Hurewicz, Menger and Rothberger,Fund. Math. 129 (1988), 17–33.[5] Just W., Miller A.W., Scheepers M., Szeptycki P.J., The combinatorics of open covers(II), Topology Appl. 73 (1996), 241–266.[6] Fremlin D., Universally Kuratowski–Ulam spaces, a note from: http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm[7] Kucharski A., Plewik S., Game approach to universally Kuratowski–Ulam spaces,Topology Appl. 154 (2007), no. 2, 421–427.[8] Kucharski A., Plewik S., Inverse systems and I-favorable spaces, Topology Appl. 156(2008), no. 1, 110–116.[9] Kuratowski K., Ulam S., Quelques propriétés topologiques du produit combinatoire,Fund. Math. 19 (1932), 247–251.[10] Oxtoby J., Measure and category, Springer-Verlag, New York, 1971.[11] Pawlikowski J., Undetermined sets of point-open games, Fund. Math. 144 (1994), 279–285.[12] Scheepers M., Combinatorics of open covers (V): Pixley–Roy spaces of sets of reals,and ω-covers., Topology Appl. 102 (2000), no. 1, 13–31.[13] Szymański A., Some applications of tiny sequences, in: Proceedings of the 11th winterschool on abstract analysis (Železna Rudá, 1983), Rend. Circ. Mat. Palermo (2) 1984,Suppl. no. 3, pp. 321–328.[14] Tsaban B., Strong γ-sets and other singular spases, Topology Appl. 153 (2005), 620–639.Institute of MathematicsUniversity of SilesiaBankowa 1440-007 KatowicePolande-mail: piotr.kalemba@us.edu.ple-mail: akuchar@math.us.edu.pl

Annales Mathematicae Silesianae

Universally Kuratowski-Ulam spaces and open-open games

Abstract
               We examine the class of spaces in which the second player has a winning strategy in the open-open game. We show that this spaces are not universally Kuratowski–Ulam. We also show that the games G and G
                  7 introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205–220] are not equivalent.</jats:p

Piotr Kalemba

Andrzej Kucharski

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Universally Kuratowski–Ulam Spaces And Open-Open Games

Universally Kuratowski-Ulam spaces and open-open games

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