Using the construction of Containing Spaces given in [1] we define some kind of games considered on topological classes of spaces

Georgiou, D. N.

Iliadis, S. D.

English

ESE - Salento University Publishing

Note di Matematica 27, suppl. n. 1, 2007, 89–96.Games on classes of spacesD. N. GeorgiouiDepartment of Mathematics, University of Patras,265 04 Patras, Greecegeorgiou@math.upatras.grS. D. IliadisiDepartment of Mathematics, University of Patras,265 04 Patras, Greeceiliadis@math.upatras.grReceived: 09/06/2006; accepted: 07/09/2006.Abstract. Using the construction of Containing Spaces given in [1] we define some kind ofgames considered on topological classes of spaces.Keywords: topological game, containing space, saturated class of spacesMSC 2000 classification: primary 54B99, 54C25, 90D44AgreementIn the present paper we denote by τ a ﬁxed inﬁnite cardinal. The ﬁrst inﬁnitecardinal larger than τ is denoted by τ+. By ω we denote the ﬁrst inﬁnite cardinal.An ordinal α is identiﬁed with the set of all ordinals less than α. A cardinalis identiﬁed with the corresponding initial ordinal. The cardinality of a set Xis denoted by |X|. By F we denote the set of all ﬁnite subsets (including theempty set) of τ .An arbitrary considered space is assumed to be a T0-space of weight lessthan or equal to τ .Let S be an indexed collection of spaces. In [1] a poset C(S), whose orderis denoted by ≺S, is constructed. This poset is directed and each subset A ofC(S) of cardinality ≤ τ has supremum denoted by sup(A). (Therefore C(S) isdirected and τ -complete) Also, for each element c ∈ C(S) a topological space,denoted by T(c) and called a Containing Space, is constructed. The purpose ofthe introduction of these notions is the construction of some special spectrumin the “theory of Containing Spaces”.We shall use the above notions to introduce games on classes of spaces. WeiThe research is funded by the European Social Fund (ESF), Operational Program for Edu-cational and Vocational Training II (EPEAEK II), and particularly the Program PYTHAGO-RAS II.90 D. N. Georgiou, S. D. Iliadisstart given some necessary notions and notations of [1] in order to explain brieﬂythe constructions of C(S) and T(c).1 The constructions of the poset C(S) and spacesT(c), c ∈ C(S)An indexed collectionM ≡ {{UXδ : δ ∈ τ } : X ∈ S }, (1)where {UXδ : δ ∈ τ } is an indexed base for X, is called a co-mark of S.A co-mark (1) of S is called a co-extension of a co-markM+ ≡ {{V Xδ : δ ∈ τ } : X ∈ S }of S if there exists an one-to-one mapping θ of τ into itself such that V Xδ = UXθ(δ),for every X ∈ S and δ ∈ τ .Consider the co-mark (1) of S. We deﬁne a familyRM ≡ {∼sM: s ∈ F }of equivalence relations on S as follows: two elements X and Y of S are ∼sM-equivalent if and only if there exists an isomorphism i of the algebra of subsetsof X generated by the ﬁnite set {UXδ : δ ∈ s } of X onto the algebra of subsetsof Y generated by the ﬁnite set {UYδ : δ ∈ s } of Y such that i(UXδ ) = UYδ ,δ ∈ s.An indexed family R ≡ {∼s: s ∈ F } of equivalence relations on S is said tobe admissible if the following conditions are satisﬁed:(a) ∼t ⊂ ∼s, if s ⊂ t ∈ F ,(b) the number of ∼s-equivalence classes is ﬁnite for every s ∈ F , and(c) ∼s= S× S if s = ∅.The set of all ∼s-equivalence classes for all s ∈ F , is denoted by C(R). By C♦(R)we denote the ring of subsets of S generated by C(R).Note that the family RM is admissible.LetR0 ≡ {∼s0: s ∈ F }andR1 ≡ {∼s1: s ∈ F }Games on classes of spaces 91be two indexed families of equivalence relations on S. It is said that R1 is a finalrefinement of R0 if for every s ∈ F there exists an element t ∈ F such that∼t1⊂∼s0.A family R of equivalence relations on S is called M-admissible if R is ﬁnalreﬁnement of RM.Let R ≡ {∼s: s ∈ F } be anM-admissible family of equivalence relations onS. On the set of all pairs (x,X), where x ∈ X ∈ S, we consider an equivalencerelation, denoted by ∼MR , as follows: (x,X) ∼MR (y, Y ) if and only if X ∼s Yfor every s ∈ F , and either x ∈ UXδ and y ∈ UYδ or x /∈ UXδ and y /∈ UYδ forevery δ ∈ τ . The set of all equivalence classes of the relation ∼MR is denotedby T ≡ T(M,R). For every s ∈ F , an equivalence class H of ∼s, and δ ∈ τwe denote by UTδ (H) (respectively, by T(M,R,H)) the set of all elements aof T(M,R) such that there exists a pair (x,X) of a for which x ∈ UXδ andX ∈ H (respectively, for which X ∈ H). The set of all UTδ (H) is denoted byBT. This set is a base for a topology on the set T and the corresponding spaceis a T0-space of weight ≤ τ .For every element X of S there exists a natural embedding iXT of X into thespace T deﬁned as follows: for every x ∈ X, iXT (x) = a, where a is the elementof T containing the pair (x,X). Thus, we have constructed Containing SpacesT(M,R) for S of weight ≤ τ .A class IP of spaces is said to be saturated if for every indexed collectionS of elements of IP there exists a co-mark M+ of S satisfying the followingcondition: for every co-extensionM ofM+ there exists anM-admissible familyR+ of equivalence relations on S such that T(M,R,H) ∈ IP for every admissiblefamily R, which is ﬁnal reﬁnement of R+, and H ∈ C♦(R). The above co-markM+ is called an initial co-mark of S corresponding to the class IP and the familyR+ is called an initial family of S corresponding to the co-markM and the classIP .Now, we give the construction of C(S). Any indexed collection{UX : X ∈ S },where UX is an open subset of X ∈ S, is called an S-open set. Let M ={ {UXε : ε ∈ τ } : X ∈ S } be a co-mark of S. For every ε ∈ τ the S-open set{UXε : X ∈ S } is called the ε-component (or a component) of M.On the set of all co-marks of S we deﬁne a partial order, denoted by ≺cm, asfollows. For two co-marks M0 and M1 we write M0 ≺cm M1 if for every δ ∈ τthere exists an element ε of τ such that the δ-component of M0 coincides withthe ε-component of M1. We note that if M1 is a co-extension of M0 of S, thenM0 ≺cm M1.By P(S) we denote the set of all pairs (M,R), where M is a co-mark of Sand R is anM-admissible family of equivalence relations on S. On the set P(S)92 D. N. Georgiou, S. D. Iliadiswe deﬁne a (partial) preorder, denoted by ≺cmaf , and an equivalence relation onS, denoted by ∼cmaf , as follows. For two elements (M0,R0) and (M1,R1) of P(S)we write:(1) (M0,R0) ≺cmaf (M1,R1) if M0 ≺cm M1 and R1 is ﬁnal reﬁnement of R0.(2) (M0,R0) ∼cmaf (M1,R1) if(M0,R0) ≺cmaf (M1,R1) and (M1,R1) ≺cmaf (M0,R0).By C(S) we denote the set of all ∼cmaf -equivalence classes. On the set C(S)we deﬁne a partial order, denoted by ≺S, as follows: for two elements c0 ≡{(M0,R0)} and c1 ≡ {(M,R1)} of C(S) we write c0 ≺S c1 if and only if(M0,R0) ≺cmaf (M1,R1). The partial ordered set C(S) is directed and eachsubset A of C(S) of cardinality ≤ τ has supremum denoted by sup(A).If (M0,R0), (M1,R1) ∈ c ∈ C(S), then T(M0,R0) = T(M1,R1). By T(c)we denote the space T(M,R), where (M,R) ∈ c.For a class IP of spaces we setC(IP ) = { c ∈ C(S) : T(c) ∈ IP }.A non-empty class IP of spaces is said to be second-type saturated if for everyindexed collection S of elements of IP the set C(IP ) contains a coﬁnal τ -closedsubset of C(S) (that is the set C(IP ) contains a coﬁnal subset A such that forevery chain A′ of A with |A′| ≤ τ , we have sup(A′) ∈ A whenever the sup(A′)exists).In what follows, we denote by IP (S) the class of all spaces homeomorphic toan element of S.2 The game Gν(S)Let ν be an inﬁnite cardinal, ν ≤ τ . Players I and II take turns in playingelements of C(S) as follows:Player I: c10 c11 · · · c1δ · · ·· · ·Player II: c20 c21 · · · c2δ · · ·Note that at all limit stages Player I goes ﬁrst. So, we have two (transﬁnite)sequences (c1δ : δ ∈ ν) and (c2δ : δ ∈ ν) of elements of C(S). We say that PlayerII wins this run of the game provided that T(c) belongs to IP (S), where c is thesupremum of the set{ c1δ : δ ∈ ν } ∪ { c2δ : δ ∈ ν }.Thus, Player I wins if T(c) 6∈ IP (S). This game is denoted by Gν(S).Games on classes of spaces 933 The game Gcm(S)This game is deﬁned as follows: Players I and II take turns in playing S-opensets USδ ≡ {UXδ : X ∈ S }, δ ∈ τ , as follows:Player I: US0 US2 · · · USδ · · ·· · ·Player II: US1 US3 · · · USδ+1 · · ·Note that at all limit stages Player I goes ﬁrst.We say that player II wins this run of the game if the indexed collectionM ≡ {{UXδ : δ ∈ τ } : X ∈ S }is a co-mark of S and T(M,RM) ∈ IP (S). Thus, player I wins if eitherM is nota co-mark of S or M is a co-mark of S and T(M,RM) 6∈ IP (S).1 Proposition. Let S be an indexed collection of spaces such that IP (S) isa second type saturated class of spaces. Then, Player II has a winning strategyin the game Gν(S).Proof. Since IP (S) is a second type saturated class the set C(IP (S)) con-tains a coﬁnal τ -closed subset of C(S). Denote this subset by A and let { aδ :δ ∈ µ } be an indication of A, where µ is the cardinal of A.We deﬁne a strategy for the Player II choosing the element c2δ , δ ∈ ν, asfollows. LetCδ = { c1η : η ∈ δ } ∪ { c2η : η ∈ δ } ∪ { c1δ }.Obviously |Cδ| ≤ ν. If sup(Cδ) ∈ A, then we set c2δ = sup(Cδ). If sup(Cδ) 6∈ A,then we denote by ε the minimal of all ordinals η such that sup(Cδ) ≺S aη. Inthis case, we set c2δ = aε. Thus, c2δ ∈ A for every δ ∈ ν.Since A is τ -closed and ν ≤ τ , by the choose of elements of c2δ we havec ≡ sup({ c1δ : δ ∈ ν } ∪ { c2δ : δ ∈ ν }) = sup({ c2δ : δ ∈ ν }) ∈ A,which means that T (c) ∈ IP (S). Thus, the deﬁned strategy for the Player II iswinning. QED2 Proposition. Let S be an indexed collection of spaces such that IP (S) isa second type saturated class of spaces. Then, Player II has winning strategy inthe game Gcm(IP (S)).Proof. Since IP (S) is a second type saturated class there exists a set Awhich is a coﬁnal τ -closed subset of C(S) such that T (c) ∈ IP (S) for everyc ∈ A.94 D. N. Georgiou, S. D. IliadisFor every S-open set US we denote by c(US) an element of A and by(M(US),R(US)) an element of c(US) such that US is the ε-component ofM(US)for some ε ∈ τ . In what follows, we shall identify the S-open set US with thesequence s ≡ (US0 ) consisting of one element.By induction, for every (transﬁnite) sequence s ≡ (US0 , US1 , . . . , USδ ) of S-open sets of the ordinal-type δ + 1, where δ ∈ τ \ { 0 }, we denote by c(s) anelement of A and by (M(s),R(s)) an element of c(s) such that: (a) for everyδ′ ∈ δ we have(M(s′),R(s′)) ≺cmaf (M(s),RM(s)) ≺cmaf (M(s),R(s)),where s′ ≡ (US0 , US1 , . . . , USδ′), (therefore, c(s′) ≺S c(s)), and (b) USδ is the ε-component of M(s) for some ε ∈ τ . Note that the existence of a co-mark M(s)satisfying relation(M(s′),R(s′)) ≺cmaf (M(s),RM(s))follows by Lemma 8.2.6 of [1].We denote by ψ a one-to-one map of τ × τ into τ satisfying the followingconditions: (a) the subset ψ(τ × τ) coincides with the set of all odd ordinal ofτ (the limits ordinal are considered to be even) and (b) for every odd ordinal δof τ , ψ−1(δ) is an element of the set τ × { δ′ } for some δ′ ∈ δ. The existence ofsuch a map is clear.Now, we deﬁne a strategy for the Player II. This means that for each oddordinal δ = δ0 + 1 the (transﬁnite) sequence sδ0 ≡ (US0 , . . . , USδ0) must uniquelydetermine the S-open set USδ . For this purpose, this set is chosen by inductionas follows. If ψ−1(δ) = (ε, η), then the sequence sη ≡ (US0 , . . . , USη ) is alreadydeﬁned and, therefore, is deﬁned the pair (M(sη),R(sη)). LetM(sη) = { {Uη,Xκ : κ ∈ τ } : X ∈ S }.Then, we set USδ = {Uη,Xε : X ∈ S }.Now, we prove that the deﬁned strategy for the Player II is winning. Thatis, if USδ = {UXδ : X ∈ S }, δ ∈ τ , then the indexed collectionM ≡ {{UXδ : δ ∈ τ } : X ∈ S }is a co-mark of S and T(M,RM) ∈ IP (S). Note that USδ is the δ-component ofM.First, we prove that for every η ∈ τ , any component ofM(sη) is a componentof M. Indeed, let ε ∈ τ . There exists an odd ordinal δ ∈ τ such that ψ−1(δ) =(ε, η). By constructionUSδ = {Uη,Xε : X ∈ S }.Games on classes of spaces 95This means that the ε-component of M(sη) is a δ-component of M. Thus, M isa co-mark of S and M(sη) ≺cm M for η ∈ δ. The last relation implies that(M(sη),RM(sη)) ≺cmaf (M,RM)and, therefore,{(M(sη),RM(sη))} ≺S {(M,RM)}, η ∈ τ.Now, suppose that there exists {(M′,R′)} ∈ C(S) such that{(M(sη),R(sη))} ≺S {(M′,R′)}.Then,(M(sη),RM(sη)) ≺cmaf (M′,R′).In particular, this imply that any component ofM(sη) is a component ofM′ and,therefore, any component ofM is a component ofM′, which meansM ≺cm M′.Thus, in order to prove that{(M,RM)} ≺S {(M′,R′)}it is suﬃces to prove that R′ is a ﬁnal reﬁnement of RM. Suppose thatRM = {∼sM: s ∈ F },RM′ = {∼sM′ : s ∈ F },R(sη) = {∼sη: s ∈ F }, η ∈ τ,andR′ = {∼′,s: s ∈ F }.Let s ≡ { δ0, . . . , δn } ∈ F . For every i ∈ { 0, . . . , n }, there exist εi, ηi ∈ τsuch that εi-component ofM(sηi) coincides with δi-component ofM. Also, thereexist ε′i such that ε′i-component of M′ coincides with δi-component of M. Lett′ = { ε′0, . . . , ε′n } ∈ F . Since R′ is ﬁnal reﬁnement of RM′ , there exists t ∈ Fsuch that ∼′,t⊆∼t′M′. On the other hand, ∼sM=∼t′M′and, therefore, ∼′,tM′⊆∼sM,which means that R′ is ﬁnal reﬁnement of RM. Hence,{(M,RM)} ≺S {(M′,R′)}which means thatc = {(M,RM)} = sup{ {(M(sη),R(sη))} : η ∈ δ }.Since A is τ -closed, c ∈ A and, therefore, T (c) ∈ IP (S). Thus, the deﬁnedstrategy of Player II is winning. QED96 D. N. Georgiou, S. D. Iliadis3 Corollary. Let S be an indexed collection of spaces such that IP (S) is oneof the following classes of spaces:(1) The class of all spaces.(2) The class of countably-dimensional spaces.(3) The class of all strongly countable dimensional spaces.(4) The class of all locally finite-dimensional spaces.(5) The class of all spaces of dimension ind ≤ α, where α ∈ τ+.Then, Player II has a winning strategy in the games Gν(S) and Gcm(S).Proof. The proof of this corollary follows by the fact that the mentionedclasses of spaces are second type saturated classes (see [1]). QED4 Problem. Let S be an indexed collection of spaces such that Player IIhas a winning strategy in the game Gν(S) (respectively, in the game Gcm(S)).Is the class IP (S) second type saturated?In our opinion the answer to Problem 1 is negative. In this case, it is inter-esting to study topological properties of the class IP (S). One of such propertyis the existence of universal elements. We have the following proposition.5 Proposition. Let S be an indexed collection of spaces such that Player IIhas a winning strategy in the game Gν(S) (respectively, in the game Gcm(S)).Then, the class IP (S) of spaces has a universal element.Proof. Since the Player II has a winning strategy in the game Gν(S) (re-spectively, in the game Gcm(S)) there exists an element c ∈ C(S) (respectively,a co-markM ≡ {{UXδ : δ ∈ τ } : X ∈ S }of S) such that T(c) ∈ IP (S) (respectively, T(M,RM) ∈ IP (S)). Since T(c) andT(M,RM) are Containing Spaces, each element X of S and, therefore, eachelement X of IP (S) is contained topologically in these spaces, which means thatthey are universal elements in IP (S). QEDReferences[1] S. D. Iliadis: Universal spaces and mappings. North-Holland Mathematics Studies, 198.Elsevier Science B.V., Amsterdam 2005. pp. xvi+559

Games on classes of spaces

Università del Salento: ESE - Salento University Publishing

Note di Matematica 27, suppl. n. 1, 2007, 89–96.Games on classes of spacesD. N. GeorgiouiDepartment of Mathematics, University of Patras,265 04 Patras, Greecegeorgiou@math.upatras.grS. D. IliadisiDepartment of Mathematics, University of Patras,265 04 Patras, Greeceiliadis@math.upatras.grReceived: 09/06/2006; accepted: 07/09/2006.Abstract. Using the construction of Containing Spaces given in [1] we define some kind ofgames considered on topological classes of spaces.Keywords: topological game, containing space, saturated class of spacesMSC 2000 classification: primary 54B99, 54C25, 90D44AgreementIn the present paper we denote by τ a fixed infinite cardinal. The first infinitecardinal larger than τ is denoted by τ+. By ω we denote the first infinite cardinal.An ordinal α is identified with the set of all ordinals less than α. A cardinalis identified with the corresponding initial ordinal. The cardinality of a set Xis denoted by |X|. By F we denote the set of all finite subsets (including theempty set) of τ .An arbitrary considered space is assumed to be a T0-space of weight lessthan or equal to τ .Let S be an indexed collection of spaces. In [1] a poset C(S), whose orderis denoted by ≺S, is constructed. This poset is directed and each subset A ofC(S) of cardinality ≤ τ has supremum denoted by sup(A). (Therefore C(S) isdirected and τ -complete) Also, for each element c ∈ C(S) a topological space,denoted by T(c) and called a Containing Space, is constructed. The purpose ofthe introduction of these notions is the construction of some special spectrumin the “theory of Containing Spaces”.We shall use the above notions to introduce games on classes of spaces. WeiThe research is funded by the European Social Fund (ESF), Operational Program for Edu-cational and Vocational Training II (EPEAEK II), and particularly the Program PYTHAGO-RAS II.90 D. N. Georgiou, S. D. Iliadisstart given some necessary notions and notations of [1] in order to explain brieflythe constructions of C(S) and T(c).1 The constructions of the poset C(S) and spacesT(c), c ∈ C(S)An indexed collectionM ≡ {{UXδ : δ ∈ τ } : X ∈ S }, (1)where {UXδ : δ ∈ τ } is an indexed base for X, is called a co-mark of S.A co-mark (1) of S is called a co-extension of a co-markM+ ≡ {{V Xδ : δ ∈ τ } : X ∈ S }of S if there exists an one-to-one mapping θ of τ into itself such that V Xδ = UXθ(δ),for every X ∈ S and δ ∈ τ .Consider the co-mark (1) of S. We define a familyRM ≡ {∼sM: s ∈ F }of equivalence relations on S as follows: two elements X and Y of S are ∼sM-equivalent if and only if there exists an isomorphism i of the algebra of subsetsof X generated by the finite set {UXδ : δ ∈ s } of X onto the algebra of subsetsof Y generated by the finite set {UYδ : δ ∈ s } of Y such that i(UXδ ) = UYδ ,δ ∈ s.An indexed family R ≡ {∼s: s ∈ F } of equivalence relations on S is said tobe admissible if the following conditions are satisfied:(a) ∼t ⊂ ∼s, if s ⊂ t ∈ F ,(b) the number of ∼s-equivalence classes is finite for every s ∈ F , and(c) ∼s= S× S if s = ∅.The set of all ∼s-equivalence classes for all s ∈ F , is denoted by C(R). By C♦(R)we denote the ring of subsets of S generated by C(R).Note that the family RM is admissible.LetR0 ≡ {∼s0: s ∈ F }andR1 ≡ {∼s1: s ∈ F }Games on classes of spaces 91be two indexed families of equivalence relations on S. It is said that R1 is a finalrefinement of R0 if for every s ∈ F there exists an element t ∈ F such that∼t1⊂∼s0.A family R of equivalence relations on S is called M-admissible if R is finalrefinement of RM.Let R ≡ {∼s: s ∈ F } be anM-admissible family of equivalence relations onS. On the set of all pairs (x,X), where x ∈ X ∈ S, we consider an equivalencerelation, denoted by ∼MR , as follows: (x,X) ∼MR (y, Y ) if and only if X ∼s Yfor every s ∈ F , and either x ∈ UXδ and y ∈ UYδ or x /∈ UXδ and y /∈ UYδ forevery δ ∈ τ . The set of all equivalence classes of the relation ∼MR is denotedby T ≡ T(M,R). For every s ∈ F , an equivalence class H of ∼s, and δ ∈ τwe denote by UTδ (H) (respectively, by T(M,R,H)) the set of all elements aof T(M,R) such that there exists a pair (x,X) of a for which x ∈ UXδ andX ∈ H (respectively, for which X ∈ H). The set of all UTδ (H) is denoted byBT. This set is a base for a topology on the set T and the corresponding spaceis a T0-space of weight ≤ τ .For every element X of S there exists a natural embedding iXT of X into thespace T defined as follows: for every x ∈ X, iXT (x) = a, where a is the elementof T containing the pair (x,X). Thus, we have constructed Containing SpacesT(M,R) for S of weight ≤ τ .A class IP of spaces is said to be saturated if for every indexed collectionS of elements of IP there exists a co-mark M+ of S satisfying the followingcondition: for every co-extensionM ofM+ there exists anM-admissible familyR+ of equivalence relations on S such that T(M,R,H) ∈ IP for every admissiblefamily R, which is final refinement of R+, and H ∈ C♦(R). The above co-markM+ is called an initial co-mark of S corresponding to the class IP and the familyR+ is called an initial family of S corresponding to the co-markM and the classIP .Now, we give the construction of C(S). Any indexed collection{UX : X ∈ S },where UX is an open subset of X ∈ S, is called an S-open set. Let M ={ {UXε : ε ∈ τ } : X ∈ S } be a co-mark of S. For every ε ∈ τ the S-open set{UXε : X ∈ S } is called the ε-component (or a component) of M.On the set of all co-marks of S we define a partial order, denoted by ≺cm, asfollows. For two co-marks M0 and M1 we write M0 ≺cm M1 if for every δ ∈ τthere exists an element ε of τ such that the δ-component of M0 coincides withthe ε-component of M1. We note that if M1 is a co-extension of M0 of S, thenM0 ≺cm M1.By P(S) we denote the set of all pairs (M,R), where M is a co-mark of Sand R is anM-admissible family of equivalence relations on S. On the set P(S)92 D. N. Georgiou, S. D. Iliadiswe define a (partial) preorder, denoted by ≺cmaf , and an equivalence relation onS, denoted by ∼cmaf , as follows. For two elements (M0,R0) and (M1,R1) of P(S)we write:(1) (M0,R0) ≺cmaf (M1,R1) if M0 ≺cm M1 and R1 is final refinement of R0.(2) (M0,R0) ∼cmaf (M1,R1) if(M0,R0) ≺cmaf (M1,R1) and (M1,R1) ≺cmaf (M0,R0).By C(S) we denote the set of all ∼cmaf -equivalence classes. On the set C(S)we define a partial order, denoted by ≺S, as follows: for two elements c0 ≡{(M0,R0)} and c1 ≡ {(M,R1)} of C(S) we write c0 ≺S c1 if and only if(M0,R0) ≺cmaf (M1,R1). The partial ordered set C(S) is directed and eachsubset A of C(S) of cardinality ≤ τ has supremum denoted by sup(A).If (M0,R0), (M1,R1) ∈ c ∈ C(S), then T(M0,R0) = T(M1,R1). By T(c)we denote the space T(M,R), where (M,R) ∈ c.For a class IP of spaces we setC(IP ) = { c ∈ C(S) : T(c) ∈ IP }.A non-empty class IP of spaces is said to be second-type saturated if for everyindexed collection S of elements of IP the set C(IP ) contains a cofinal τ -closedsubset of C(S) (that is the set C(IP ) contains a cofinal subset A such that forevery chain A′ of A with |A′| ≤ τ , we have sup(A′) ∈ A whenever the sup(A′)exists).In what follows, we denote by IP (S) the class of all spaces homeomorphic toan element of S.2 The game Gν(S)Let ν be an infinite cardinal, ν ≤ τ . Players I and II take turns in playingelements of C(S) as follows:Player I: c10 c11 · · · c1δ · · ·· · ·Player II: c20 c21 · · · c2δ · · ·Note that at all limit stages Player I goes first. So, we have two (transfinite)sequences (c1δ : δ ∈ ν) and (c2δ : δ ∈ ν) of elements of C(S). We say that PlayerII wins this run of the game provided that T(c) belongs to IP (S), where c is thesupremum of the set{ c1δ : δ ∈ ν } ∪ { c2δ : δ ∈ ν }.Thus, Player I wins if T(c) 6∈ IP (S). This game is denoted by Gν(S).Games on classes of spaces 933 The game Gcm(S)This game is defined as follows: Players I and II take turns in playing S-opensets USδ ≡ {UXδ : X ∈ S }, δ ∈ τ , as follows:Player I: US0 US2 · · · USδ · · ·· · ·Player II: US1 US3 · · · USδ+1 · · ·Note that at all limit stages Player I goes first.We say that player II wins this run of the game if the indexed collectionM ≡ {{UXδ : δ ∈ τ } : X ∈ S }is a co-mark of S and T(M,RM) ∈ IP (S). Thus, player I wins if eitherM is nota co-mark of S or M is a co-mark of S and T(M,RM) 6∈ IP (S).1 Proposition. Let S be an indexed collection of spaces such that IP (S) isa second type saturated class of spaces. Then, Player II has a winning strategyin the game Gν(S).Proof. Since IP (S) is a second type saturated class the set C(IP (S)) con-tains a cofinal τ -closed subset of C(S). Denote this subset by A and let { aδ :δ ∈ µ } be an indication of A, where µ is the cardinal of A.We define a strategy for the Player II choosing the element c2δ , δ ∈ ν, asfollows. LetCδ = { c1η : η ∈ δ } ∪ { c2η : η ∈ δ } ∪ { c1δ }.Obviously |Cδ| ≤ ν. If sup(Cδ) ∈ A, then we set c2δ = sup(Cδ). If sup(Cδ) 6∈ A,then we denote by ε the minimal of all ordinals η such that sup(Cδ) ≺S aη. Inthis case, we set c2δ = aε. Thus, c2δ ∈ A for every δ ∈ ν.Since A is τ -closed and ν ≤ τ , by the choose of elements of c2δ we havec ≡ sup({ c1δ : δ ∈ ν } ∪ { c2δ : δ ∈ ν }) = sup({ c2δ : δ ∈ ν }) ∈ A,which means that T (c) ∈ IP (S). Thus, the defined strategy for the Player II iswinning. QED2 Proposition. Let S be an indexed collection of spaces such that IP (S) isa second type saturated class of spaces. Then, Player II has winning strategy inthe game Gcm(IP (S)).Proof. Since IP (S) is a second type saturated class there exists a set Awhich is a cofinal τ -closed subset of C(S) such that T (c) ∈ IP (S) for everyc ∈ A.94 D. N. Georgiou, S. D. IliadisFor every S-open set US we denote by c(US) an element of A and by(M(US),R(US)) an element of c(US) such that US is the ε-component ofM(US)for some ε ∈ τ . In what follows, we shall identify the S-open set US with thesequence s ≡ (US0 ) consisting of one element.By induction, for every (transfinite) sequence s ≡ (US0 , US1 , . . . , USδ ) of S-open sets of the ordinal-type δ + 1, where δ ∈ τ \ { 0 }, we denote by c(s) anelement of A and by (M(s),R(s)) an element of c(s) such that: (a) for everyδ′ ∈ δ we have(M(s′),R(s′)) ≺cmaf (M(s),RM(s)) ≺cmaf (M(s),R(s)),where s′ ≡ (US0 , US1 , . . . , USδ′), (therefore, c(s′) ≺S c(s)), and (b) USδ is the ε-component of M(s) for some ε ∈ τ . Note that the existence of a co-mark M(s)satisfying relation(M(s′),R(s′)) ≺cmaf (M(s),RM(s))follows by Lemma 8.2.6 of [1].We denote by ψ a one-to-one map of τ × τ into τ satisfying the followingconditions: (a) the subset ψ(τ × τ) coincides with the set of all odd ordinal ofτ (the limits ordinal are considered to be even) and (b) for every odd ordinal δof τ , ψ−1(δ) is an element of the set τ × { δ′ } for some δ′ ∈ δ. The existence ofsuch a map is clear.Now, we define a strategy for the Player II. This means that for each oddordinal δ = δ0 + 1 the (transfinite) sequence sδ0 ≡ (US0 , . . . , USδ0) must uniquelydetermine the S-open set USδ . For this purpose, this set is chosen by inductionas follows. If ψ−1(δ) = (ε, η), then the sequence sη ≡ (US0 , . . . , USη ) is alreadydefined and, therefore, is defined the pair (M(sη),R(sη)). LetM(sη) = { {Uη,Xκ : κ ∈ τ } : X ∈ S }.Then, we set USδ = {Uη,Xε : X ∈ S }.Now, we prove that the defined strategy for the Player II is winning. Thatis, if USδ = {UXδ : X ∈ S }, δ ∈ τ , then the indexed collectionM ≡ {{UXδ : δ ∈ τ } : X ∈ S }is a co-mark of S and T(M,RM) ∈ IP (S). Note that USδ is the δ-component ofM.First, we prove that for every η ∈ τ , any component ofM(sη) is a componentof M. Indeed, let ε ∈ τ . There exists an odd ordinal δ ∈ τ such that ψ−1(δ) =(ε, η). By constructionUSδ = {Uη,Xε : X ∈ S }.Games on classes of spaces 95This means that the ε-component of M(sη) is a δ-component of M. Thus, M isa co-mark of S and M(sη) ≺cm M for η ∈ δ. The last relation implies that(M(sη),RM(sη)) ≺cmaf (M,RM)and, therefore,{(M(sη),RM(sη))} ≺S {(M,RM)}, η ∈ τ.Now, suppose that there exists {(M′,R′)} ∈ C(S) such that{(M(sη),R(sη))} ≺S {(M′,R′)}.Then,(M(sη),RM(sη)) ≺cmaf (M′,R′).In particular, this imply that any component ofM(sη) is a component ofM′ and,therefore, any component ofM is a component ofM′, which meansM ≺cm M′.Thus, in order to prove that{(M,RM)} ≺S {(M′,R′)}it is suffices to prove that R′ is a final refinement of RM. Suppose thatRM = {∼sM: s ∈ F },RM′ = {∼sM′ : s ∈ F },R(sη) = {∼sη: s ∈ F }, η ∈ τ,andR′ = {∼′,s: s ∈ F }.Let s ≡ { δ0, . . . , δn } ∈ F . For every i ∈ { 0, . . . , n }, there exist εi, ηi ∈ τsuch that εi-component ofM(sηi) coincides with δi-component ofM. Also, thereexist ε′i such that ε′i-component of M′ coincides with δi-component of M. Lett′ = { ε′0, . . . , ε′n } ∈ F . Since R′ is final refinement of RM′ , there exists t ∈ Fsuch that ∼′,t⊆∼t′M′. On the other hand, ∼sM=∼t′M′and, therefore, ∼′,tM′⊆∼sM,which means that R′ is final refinement of RM. Hence,{(M,RM)} ≺S {(M′,R′)}which means thatc = {(M,RM)} = sup{ {(M(sη),R(sη))} : η ∈ δ }.Since A is τ -closed, c ∈ A and, therefore, T (c) ∈ IP (S). Thus, the definedstrategy of Player II is winning. QED96 D. N. Georgiou, S. D. Iliadis3 Corollary. Let S be an indexed collection of spaces such that IP (S) is oneof the following classes of spaces:(1) The class of all spaces.(2) The class of countably-dimensional spaces.(3) The class of all strongly countable dimensional spaces.(4) The class of all locally finite-dimensional spaces.(5) The class of all spaces of dimension ind ≤ α, where α ∈ τ+.Then, Player II has a winning strategy in the games Gν(S) and Gcm(S).Proof. The proof of this corollary follows by the fact that the mentionedclasses of spaces are second type saturated classes (see [1]). QED4 Problem. Let S be an indexed collection of spaces such that Player IIhas a winning strategy in the game Gν(S) (respectively, in the game Gcm(S)).Is the class IP (S) second type saturated?In our opinion the answer to Problem 1 is negative. In this case, it is inter-esting to study topological properties of the class IP (S). One of such propertyis the existence of universal elements. We have the following proposition.5 Proposition. Let S be an indexed collection of spaces such that Player IIhas a winning strategy in the game Gν(S) (respectively, in the game Gcm(S)).Then, the class IP (S) of spaces has a universal element.Proof. Since the Player II has a winning strategy in the game Gν(S) (re-spectively, in the game Gcm(S)) there exists an element c ∈ C(S) (respectively,a co-markM ≡ {{UXδ : δ ∈ τ } : X ∈ S }of S) such that T(c) ∈ IP (S) (respectively, T(M,RM) ∈ IP (S)). Since T(c) andT(M,RM) are Containing Spaces, each element X of S and, therefore, eachelement X of IP (S) is contained topologically in these spaces, which means thatthey are universal elements in IP (S). QEDReferences[1] S. D. Iliadis: Universal spaces and mappings. North-Holland Mathematics Studies, 198.Elsevier Science B.V., Amsterdam 2005. pp. xvi+559

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Games on classes of spaces

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