Topological Games and Covering Dimension

Transcription

1 Volume 38, 2011 Pages Topological Games and Covering Dimension by Liljana Babinkostova Electronically published on August 16, 2010 Topology Proceedings Web: Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume 38 (2011) Pages E-Published on August 16, 2010 TOPOLOGICAL GAMES AND COVERING DIMENSION LILJANA BABINKOSTOVA Abstract. We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional separable metric spaces. All spaces in this paper are assumed to be separable metric spaces. Infinite games can be used in a natural way to define ordinal valued functions on the class of separable metric spaces. One of our examples of such an ordinal function coincides in any finite dimensional metric spaces with the covering dimension of the space, and may thus be thought of as an extension of Lebesgue covering dimension to all separable metric spaces. We will call this particular extension of Lebesgue covering dimension the game dimension of a space. Game dimension is defined using a game motivated by a selection principle. Several natural classical selection principles are related to the one motivating game dimension, and their associated games can be used to compute upper bounds on game dimension. These games, and the upper bounds they provide, are interesting in their own right. We develop two such games and use them then to obtain upper bounds on our game dimension Mathematics Subject Classification. Primary 54D20, 54D45, 55M10; Secondary 03E20. Key words and phrases. countable dimensional, game dimension, infinite game, selection principle. c 2010 Topology Proceedings. 99

3 100 L. BABINKOSTOVA 1. Selection principles, open covers, and games The selection principle S 1 (A, B) states that there is for each sequence (A n : n N) of elements of A a sequence (B n : n N) such that for each n, we have B n A n, and {B n : n N} B. The selection principle S c (A, B) states that there is for each sequence (A n : n N) of members of A a sequence (B n : n N) of sets such that for each n, B n is a pairwise disjoint family of sets and refines A n and n N B n B. For a collection T of subsets of a topological space X, we call an open cover U of X a T -cover if, for each T T, there is a U U with T U. The symbol O(T ) denotes the collection of T -covers of X. A trivial situation, but one we cannot ignore, arises when X itself is a member of T. We don t follow the usual practice of requiring X U for U O(T ). The motivation, as will be seen below, is that allowing this trivial situation provides a uniformity to the statements of some of our results. A few special combinatorial properties of the family T are important for our considerations. Here are some of them: T is said to be up-directed if for all A and B in T there is a C in T with A B C. T is said to be first-countable if there is for each T T a sequence (U n : n N) of open sets such that for each n, U n U n+1 T, and for each open set V T, there is an n with V U n. We shall say that X is T -first countable if the family T is first countable. Let T denote the set of subspaces which are unions of countably many elements of T. When T is a collection of compact sets in a metrizable space X, T is first countable and up-directed. Call a subset C of T cofinal if there is for each T T a C C with T C. When T is the collection of one-element subsets of X, then O(T ) will be denoted simply O. Let FD denote the collection of finite dimensional subsets of X, and let KFD denote the collection of compact, finite dimensional subsets of X. Then O fd denotes O(FD) and O kfd denotes O(KFD). When T is the collection of finite subsets of X, then T -covers are called ω-covers, and Ω denotes O(T ). The collection of open covers having just two elements is denoted O 2. A topological space is said to be weakly infinite dimensional if it has the property S c (O 2, O). The class of spaces satisfying S c (O, O) was introduced in [1]. If a space is a union of countably

4 TOPOLOGICAL GAMES AND COVERING DIMENSION 101 many zero-dimensional subsets, it is said to be countable dimensional. Hurewicz introduced the latter notion. As was shown in [1], countable dimensional implies S c (O, O), and S c (O, O) implies S c (O 2, O). If a space is not weakly infinite dimensional, then it is said to be strongly infinite dimensional. The Hilbert cube H is an example of strongly infinite dimensional space. Let α be an ordinal number. The game G α 1 (A, B) associated with the selection principle S 1 (A, B) is as follows: The players play an inning per γ < α. In the γ-th inning, ONE first chooses an A γ A; TWO responds with a B γ A γ. A play A 0, B 0,, A γ, B γ, of length α is won by TWO if {B γ : γ < α} B. Else, ONE wins. When for a set S and families A and B there is an ordinal number α such that TWO has a winning strategy in the game G α 1 (A, B) played on S, then we define tp S 1 (A,B)(S) to be min{α : TWO has a winning strategy ing α 1 (A, B) on S}. We adopt the convention that tp S1 (A,B)( ) = 0. It is important to note that the innings are numbered starting at 0. Thus, when TWO has a winning strategy guaranteeing a win in inning 0, tp S1 (A,B)(S) = 1. The ordinal-valued function tp S1 (A,B)(S) has been studied for a few specific choices of the families A and B. Examples closely affiliated with what we will examine here can be found in [4], [6], [7], [8], [16], and [15]. The following monotonicity properties are easy to check. Lemma 1.1. For each separable metric space X, for each closed set Y X, and for families S T and A B the following hold: (1) tp S1 (O(T ),A)(X) tp S1 (O(S),A)(X). (2) tp S1 (O(T ),B)(X) tp S1 (O(T ),A)(X). (3) tp S1 (O(T ),A)(Y ) tp S1 (O(T ),A)(X). In particular, we find that tp S1 (O fd,a)(x) tp S1 (O kfd,a)(x). Some of the results from our investigation in [4] have some applications in this paper. We first recall these. In order to compare the ordinals tp Sξ (A,B)(X) for various choices ξ, A, and B, we extend Lemma 1 of [4] as follows. Lemma 1.2. Let α be an ordinal number, and let F be a strategy for TWO in the game G α 1 (O(T ), O). Then there is for each ν < α, for each sequence (O β : β < ν) from O(T ) a set C T such that

5 102 L. BABINKOSTOVA for each open set U C there is an O O(T ) with F ((O β : β < ν) (O)) = U. Theorem 1.3 ([4, Theorem 4]). Let α be an infinite ordinal and let T be up-directed. If F is any strategy for TWO in G α 1 (O(T ), O) and if X is T -first countable, then there is for each set T T a set S T such that T S and for any closed set C X S, there is an ω-length F -play such that T n=1 T n X C. O 1, T 1,, O n, T n Theorem 1.4 ([4, Theorem 5]). Let α be an infinite ordinal and let T be up-directed. If F is any strategy for TWO in G α 1 (O(T ), O) and if X is C-first countable where C T is cofinal in T, then there is for each set T T a set S C such that T S and for any closed set C X S, there is an ω-length F -play such that T n=1 T n X C. O 1, T 1,, O n, T n Lemma 1.5 ([4, Theorem 6]). If T has a cofinal subset consisting of G δ -sets, then TWO has a winning strategy in G ω 1 (O(T ), O) if and only if X is a union of countably many elements of T. Lemma 1.6 ([4, Lemma 13]). Let X and T be such that X T and T is up-directed and first-countable. If α is the least ordinal for which there is a B T such that each closed set C X B satisfies tp S1 (O(T ),O)(C) α, then tp S1 (O(T ),O)(X) = ω + α. The converse is also true Lemma 1.7. Let X and T be such that X T and T is updirected and first-countable. If tp S1 (O(T ),O)(X) = ω + α, then there is a B T such that each closed set C X B satisfies tp S1 (O(T ),O)(C) α, and α is the minimal such ordinal. Proof: For let F be a winning strategy for TWO in the game G ω+α 1 (O(T ), O). By Lemma 1.2, choose C T such that for each open set U C there is a U O(T ) with U = F (U). Since T is first countable, choose a sequence (U n : n < ω) of open sets with: For each open set U C, there is an n with U U n C, and for all n, U n U n+1. For each n, choose O n O(T ) with U n = F (O n ).

6 TOPOLOGICAL GAMES AND COVERING DIMENSION 103 Applying Lemma 1.2 to each O n, fix sets C n T such that there is for each open U C n an O O(T ) with U = F (O n, O). Since T is first countable, choose for each n a sequence (U n,m : m < ω) of open sets such that: For each open set U C n there is an m with U U n,m C n, and for all m, U n,m U n,m+1. Then choose for each m an O n,m O(T ) with U n,m = F (O n, O n,m ). Continuing like this we find for each finite sequence (n 0,, n j ) sets C n0,,n j T, and open sets U n0,,n j,t, and O n0,,n j,t O(T ), t < ω, such that for each open set U C n0,,n j, there is a t with (1) U U n0,,n j,t C n0,,n j (2) U n0,,n j,t = F (O n0, O n0,n 1,, O n0,,n j,t) (3) For all t, U n0,,n j,t U n0,,n j,t+1. Put B = τ <ω ωc τ, an element of T. Consider any closed set C X B. Since U = X C B is open, choose n 0 with C U n0 U. Then choose n 1 with C n0 U n0,n 1 U, and then n 2 with C n0,n 1 U n0,n 1,n 2 U, and so on. In this way, we obtain an ω-sequence of moves during which TWO used the strategy F, and the union of TWO s responses, say V, is a subset of U. Since F is a winning strategy for TWO in the game G ω+α 1 (O(T ), O), it follows that tp S1 (O(T ),O)(C) tp S1 (O(T ),O)(X U) tp S1 (O(T ),O)(X V ) α. If α is a limit ordinal, then for each β < α there is a closed set C X B with tp S1 (O(T ),O)(C) > β. If α = β + 1 is a successor ordinal, then there is a closed set C X B with tp S1 (O(T ),O)(C) = β + 1. Thus, for appropriately chosen C, the closed set X V witnesses that α is the minimal such ordinal. 2. The ordinal tp S1 (O kfd,o)(x). The properties of open point-covers of spaces are at the basis of Lebesgue s notion of covering dimension. The points have, in that theory, dimension zero. We will consider a more general situation of open T -covers of spaces. In this case, by analogy with point covers, we require that the notion of dimension assign the value 0 to the members of T. We feature here the specific case when T is the collection of finite dimensional compact spaces. Note that when X is a compact finite-dimensional space, tp S1 (O kfd,o)(x) = 0.

7 104 L. BABINKOSTOVA 2.1. When tp S1 (O kfd,o)(x) is countable The following proposition follows from Lemma 1.5. Proposition 2.1. For X a separable metric space the following are equivalent: (1) X is a countable union of compact finite dimensional subsets. (2) 1 < tp S1 (O kfd,o)(x) ω. We shall see in subsection 2.3 that (2) of Proposition 2.1 cannot be improved Products and unions Lemma 2.2. For X a metric space, and Y a compact finite dimensional metric space, tp S1 (O kfd,o)(x Y ) = tp S1 (O kfd,o)(x). Proof: Since X is a closed subset of X Y, we have tp S1 (O kfd,o)(x) tp S1 (O kfd,o)(x Y ). And we show that tp S1 (O kfd,o)(x Y ) tp S1 (O kfd,o)(x). Let F be a winning strategy for TWO in G α 1 (O kfd, O) played on X. TWO uses F as follows to play the game on X Y. First observe that if O is in O kfd, then there is an S(O) O kfd such that each element of S(O) is of the form U Y, and such that S(O) is a refinement of O. Define a strategy G for TWO in the game G α 1 (O kfd, O) on X Y as follows: In inning 1, when ONE plays O 1, put B 1 = {U X : U Y S(O 1 )}, a member of O kfd for X. Then compute W 1 = F (B 1 ) B 1, and choose T 1 O 1 with W 1 Y T 1. Define G(O 1 ) = T 1. Let γ < α be given, as well as a sequence (O ξ : ξ γ) of members of O kfd for X Y. For each ξ γ, define B ξ = {U X : U Y S(O ξ )}, a member of O kfd on X. Compute W γ = F (B ξ : ξ γ) B γ and then choose T γ O γ with W γ Y T γ and put G(O ξ : ξ γ) = T γ. Then G is a winning strategy for TWO in G α 1 (O kfd, O) on X Y. Next we describe the structure of those metrizable spaces X with tp S1 (O kfd,o)(x) finite.

8 TOPOLOGICAL GAMES AND COVERING DIMENSION 105 Lemma 2.3. Let X be a metric space which is not compact and finite dimensional. If tp S1 (O kfd,o)(x) is finite, then there is a nonempty finite dimensional compact set C such that for each open set U C, we have tp S1 (O kfd,o)(x U) tp S1 (O kfd,o)(x) 1, and for some open set U C, we have tp S1 (O kfd,o)(x U) = tp S1 (O kfd,o)(x) 1 Proof: Suppose not. For each compact finite dimensional C X choose an open set U(C) such that C U(C), and tp S1 (O kfd,o)(x U(C)) tp S1 (O kfd,o)(x). Then O 1 = {U(C) : C X finite dimensional compact} O kfd, and for each strategy G of TWO, tp S1 (O kfd,o)(x G(O 1 )) tp S1 (O kfd,o)(x). But this contradicts the fact that tp S1 (O kfd,o)(x) is finite. Suppose that for each compact finite dimensional C X, for each open set U C, we have tp S1 (O kfd,o)(x U) tp S1 (O kfd,o)(x) 2. Then we have that tp S1 (O kfd,o)(x) tp S1 (O kfd,o)(x) 1, which is a contradiction. Lemma 2.4. For X a metric space, and Y 1,, Y n subspaces such that for each n, tp S1 (O kfd,o)(y n ) α, also tp S1 (O kfd,o)( j n Y j) α. Proof: For each n, we prove this by induction on α. For n = 1, there is nothing to prove. Thus, assume n > 1. When α = 0 there is nothing to prove since each Y j then is compact and finite dimensional, as is their union. Thus, assume that 0 < α and we have proven this result for all β < α. There are three cases to consider: α is finite, α is an infinite limit ordinal, and α is an infinite successor ordinal. When α is finite: Since we have already disposed of the case when α = 0, consider now the case where α = m + 1 and the result is proved for α m. By Lemma 2.3, choose for j n a compact finite dimensional set C j Y j such that for any open (in Y j ) set U containing C j, tp S1 (O kfd,o)(y j U) m. Then C = j n C j is compact and finite dimensional, and for every open set U C, for each j, tp S1 (O kfd,o)(y j U) m. Thus, TWO can play as follows: When ONE in inning 0 play an O 0 O kfd for j n Y j, TWO chooses T 0 O 0 with C T 0. Since now for j

9 106 L. BABINKOSTOVA n, we have for Z j = Y j T 0 that tp S1 (O kfd,o)(z j ) m, the induction hypothesis gives tp S1 (O kfd,o)( j n Z j ) m, tp S1 (O kfd,o)(( j n Y j ) T 0 ) m. It follows that tp S1 (O kfd,o)( j n Y j ) m + 1, completing the induction step and the proof for α finite. When α is a limit ordinal: We can write α = j m A j where each A j has order type α, and in innings γ A j, we use a winning strategy for TWO on Y j to choose an element T γ O γ. This plan produces a winning strategy in the α-length game on j n Y j. When α is an infinite successor ordinal: Write α = l(α) + n(α) where l(α) is a limit ordinal and n(α) < ω. During the first l(α) innings we follow the plan above. After these innings for each j, the uncovered part Z j of Y j has tp S1 (O kfd,o)(z j ) n(α), and thus the uncovered part Z of j n Y j has tp S1 (O kfd,o)(z) n(α), by the finite case. This completes the proof Examples of X for which tp S1 (O kfd,o)(x) is finite and positive Smirnov s compactum S ω is constructed as follows: For positive integer n, define S n = I n. Then define S ω to be the onepoint compactification of the topological sum n=1 S n, say S ω = ( n=1 S n) {p ω }. It is clear that S ω is a union of countably many compact finite dimensional spaces. Lemma 2.5. For each positive integer n, tp S1 (O kfd,o)(s n ω) = n + 1. Proof: The proof is by induction on n. n = 1: Since S ω is compact and not finite dimensional, TWO does not always win in one inning. When ONE plays O 0 O kfd, then TWO chooses T 0 O 0 such that p ω T 0. Then S ω T 0 is compact finite dimensional, and thus, in the next inning, TWO chooses T 1 O 1 containing this set. n = k+1: Assume that we have already proven the result for n k. Consider n = k + 1: Player ONE will play open covers O with the property that, for any element U of O 1 which contains the k+1 vector (p ω,, p ω ), there is some m > k+1 such that the projection of U in each of the k+1 directions is disjoint from j m S j. It follows that for these elements U of ONE s move, S k+1 ω U contains the

10 TOPOLOGICAL GAMES AND COVERING DIMENSION 107 closed subspace k+1 j=1 X j, where X j is the product k+1 i=1 Z i where { Sω if i = j Z i = S k+1 otherwise. Each X j is homeomorphic to S k+1 S k ω. By Lemma 2.2, we have that tp S1 (O kfd,o)(s k+1 S k ω) = tp S1 (O kfd,o)(s k ω) = k+1. Then Lemma 2.4 implies that tp S1 (O kfd,o)(s k+1 ω U) k + 1. Thus, we conclude that tp S1 (O kfd,o)(s k+1 ω ) k + 2. Proving that this inequality is, in fact, an equality is left to the reader. It follows that tp S1 (O kfd,o)( n=1 Sn ω) = ω Examples of X with tp S1 (O kfd,o)(x) ω n + 1 Pol s compactum [12] K is constructed as follows: Start with a complete hereditarily strongly infinite dimensional totally disconnected metric space M and then compactify it such that the extension L is a union of countably many compact, finite dimensional spaces. Then K = M L. For the rest of the paper fix a representation of L as a union of countably many compact finite dimensional sets, say L = n=1 L n, where, for m < n, we have L m L n and dim(l m ) < dim(l n ). Theorem 2.6. For each positive integer n, tp S1 (O kfd,o)(k n ) ω n + 1 Proof: We use induction on n. tp S1 (O kfd,o)(k) ω + 1: In inning n, when ONE plays the open cover O n O kfd, TWO chooses T n O n with L n T n. After ω innings, L n=1 T n and the part of K not yet covered by TWO is a compact set contained in the totally disconnected space M, and thus is a compact zero-dimensional set. In inning ω + 1, TWO chooses T ω O ω containing this compact zero-dimensional set. In this case, we actually have an equality since K is not countable dimensional, and by Lemma 1.5, tp S1 (O kfd,o)(k) > ω. Assume that the statement is true for k < n. tp S1 (O kfd,o)(k n ) ω n + 1: For j < n and m < ω, define K(j, m) = K j L m K n j 1. For each such (j, m), K(j, m) is homeomorphic to L m K n 1 and by Lemma 2.2 and the induction

11 108 L. BABINKOSTOVA hypothesis, tp S1 (O kfd,o)(k(j, m)) = ω (n 1) + 1. Let F j,m be TWO s winning strategy in G ω (n 1)+1 1 (O kfd, O) on K(j, m). Now write ω (n 1) as a union m<ω j<n S j,m where each S j,m has order type ω (n 1) and for (i, m) = (j, l), S i,m S j,l =. Enumerate each S j,m in an order preserving way as (s j,m γ : γ < ω (n 1)). In the first ω (n 1) innings, in inning γ, when ONE plays open cover O γ of K n, TWO responds as follows: Choose j, m with γ S j,m, and choose δ with γ = s j,m δ. Write O j,m δ for O γ. Then TWO applies the winning strategy for inning δ of the game on K(j, m) to choose T γ O γ by T γ = F j,m (Oν j,m : ν δ). Then, after ω (n 1) innings, TWO has constructed the open set V = γ<ω (n 1) T γ in such a way that for each (j, m), the set C(j, m) = K(j, m) V is compact and finite dimensional. Next enumerate {(j, m) : j < n, m < ω} bijectively as {(j k, m k ) : k < ω}. In inning ω (n 1) + k, TWO chooses T ω (n 1)+k O ω (n 1)+k containing C(j k, m k ). Put U = γ<ω n T γ. Note that M n U = K n U: For if (x 1,, x n ) M n, choose a j with x j L and choose an m with x j L m. Then (x 1,, x n ) K(j, m) and thus, if it is not in V, it is in C(j, m) and thus in U. Since M n U is compact and totally disconnected, it is compact and zero-dimensional, and so in one more inning, TWO covers M n U. In particular, for each n, K n also satisfies S 1 (O kfd, O). Corollary 2.7. For each positive integer n, n tp S1 (O kfd,o)(k n ) tp S1 (O kfd,o)(k). Lemma 2.8. Let X be metrizable with tp S1 (O kfd,o)(x) = ω. For each m, j=1 tp S1 (O kfd,o)(k m X) = tp S1 (O kfd,o)(k m )+tp S1 (O kfd,o)(x) ω (m+1). Proof: For each n, let X n be compact finite dimensional so that X = n<ω X n. We prove this theorem by induction on m. tp S1 (O kfd,o)(k X) ω 2 : By Lemma 2.2 and Theorem 2.6, for each n, TWO has a winning strategy F n in G ω+1 1 (O kfd, O) on K X n.

12 TOPOLOGICAL GAMES AND COVERING DIMENSION 109 Write ω = n< S n where each S n is infinite, and S m S n = for m = n. For each n, we enumerate S n in order as (s n j : j < ω). In the first ω innings, when ONE plays O k in inning k, TWO chooses n with k S n and then fixes j with k = s n j, and think of O k as Oj n, the j-th move of ONE in the game on K X n. Then TWO chooses T k O k using F n (O0 n,, On j ). Let U be the union of TWO s moves during the first ω innings. For each n, since TWO was using the winning strategy F n through ω innings on K X n, the set C n = (K X n ) U is compact finite dimensional. TWO covers these countably many compact finite dimensional sets in the next ω innings. Assume that m > 1 and the theorem holds for all j < m. tp S1 (O kfd,o)(k m X) ω (m + 1): By Lemma 2.2 and Theorem 2.6, for each n, TWO has a winning strategy F n in G ω m+1 1 (O kfd, O) on K m X n. Write ω m = n< S n where each S n is infinite, and S m S n = for m = n. For each n, we enumerate S n in order as (s n γ : γ < ω m). In the first ω m innings, when ONE plays O γ in inning γ, TWO chooses n with γ S n and then fixes δ with γ = s n δ, and think of O γ as Oδ n, the δ-th move of ONE in the game on Km X n. Then TWO chooses T γ O γ using F n (O0 n,, On δ ). Let U be the union of TWO s moves during the first ω m innings. For each n, since TWO was using the winning strategy F n through ω m innings on K m X n, the set C n = (K m X n ) U is compact finite dimensional. TWO covers these countably many compact finite dimensional sets in the next ω innings. Theorem 2.9. Let X be a metrizable space with tp S1 (O kfd,o)(x) a successor ordinal. Let Y be a metric space with tp S1 (O kfd,o)(y ) ω. Then tp S1 (O kfd,o)(x Y ) tp S1 (O kfd,o)(x) + ω. Proof: Write α = β + m where β(< α = tp S1 (O kfd,o)(x)) is a limit ordinal, and 0 < m < ω. Write β = n< S n where each S n is a subset of β of order type β, and S m S n = for m = n. For each n, we enumerate S n as (s n γ : γ < β). When ONE plays O γ in inning γ < β, TWO chooses n with γ S n and then fixes ν with γ = s n ν, and think of O γ as O n ν, the

13 110 L. BABINKOSTOVA ν-th move of ONE in the game on X Y n. Then TWO chooses T γ O γ using F n (O0 n,, On ν ). Let U be the union of the moves TWO made during these β innings. For each n, since TWO was using the winning strategy F n through β innings on X Y n, we have tp S1 (O kfd,o)((x Y n ) U) m. But then, by Proposition 2.1, for each n, the set C n = (X Y n ) U is a union of countably many compact finite dimensional sets. During the next ω innings, TWO covers these. Thus, we have that tp S1 (O kfd,o)(x Y ) α + ω When tp S1 (O kfd,o)(x) is a limit ordinal Theorem Let X be a metrizable space with tp S1 (O kfd,o)(x) = α a limit ordinal, and let Y be a metric space with tp S1 (O kfd,o)(y ) ω. Then tp S1 (O kfd,o)(x Y ) = α. Proof: Since tp S1 (O kfd,o)(y ) ω, by Proposition 2.1, choose compact finite dimensional sets Y n Y with Y = n<ω Y n. By Lemma 2.2, fix, for each n, a winning strategy F n for TWO in the game G α 1 (O kfd, O) played on X Y n. By Lemma 1.1(3), since X is homeomorphic to a closed subspace of X Y, we have α tp S1 (O kfd,o)(x Y ). We must show that tp S1 (O kfd,o)(x Y ) α. Write α = n< S n where each S n is a subset of α of order type α, and S m S n = for m = n. For each n, we enumerate S n in order as (s n γ : γ < α). When ONE plays O γ in inning γ < α, TWO chooses n with γ S n and then fixes ν with γ = s n ν, and think of O γ as Oν n, the ν-th move of ONE in the game on X Y n. Then TWO chooses T γ O γ using F n (O0 n,, On ν ). Let U be the union of the moves TWO made during these α innings. For each n, since TWO was using the winning strategy F n through α innings on X Y n, we have X Y n U. Thus, TWO won in α innings. This completes the proof. Let X be n=1 Kn, the topological sum of the finite powers of the Pol compactum. Then ω 2 tp S1 (O kfd,o)(x). Also note that for each n, X K n is homeomorphic to m=1 Km+n which evidently has ω 2 tp S1 (O kfd,o)(x K n ). This is a locally compact space. Let

14 TOPOLOGICAL GAMES AND COVERING DIMENSION 111 K ω = ( n=1 Kn ) {p ω } be the one-point compactification of X. It is easy to see that tp S1 (O kfd,o)(k ω ) ω 2. Lemma For each positive integer n, tp S1 (O kfd,o)(k ω K n ) ω 2. Proof: During the first ω n + 1 innings, cover {p ω } K n. The remaining part of the space is a closed subset of X K n and thus requires at most ω 2 more innings. Thus, ω 2 = ω n ω 2 tp S1 (O kfd,o)(k ω K n ). 3. The ordinal tp Sc (O,O)(S) and the game dimension We now define an ordinal-valued function on the set of subspaces of the Hilbert cube such that this function coincides with Lebesgue covering dimension in the case of subspaces with finite covering dimension, and captures an important aspect of extending the covering dimension to subspaces that are not finite dimensional. Other important examples of such ordinal-valued functions were developed by Piet Borst [5], and Roman Pol [13]. Let α be an ordinal number. The game G α c (A, B) associated with the selection principle S c (A, B) is as follows: The players play an inning per γ < α. In the γ-th inning, ONE first chooses an A γ A; TWO then responds with a pairwise disjoint family of sets B γ that refines A γ. A play A 0, B 0,, A γ, B γ, of length α is won by TWO if n N B n B. Else, ONE wins. When for a set S and families A and B there is an ordinal number α such that TWO has a winning strategy in the game G α c (A, B) played on S, then we define tp Sc (A,B)(S) to be min{α : TWO has a winning strategy ing α c (A, B) on S}. For X a separable finite-dimensional metric space, dim(x) denotes the Lebesgue covering dimension of X. The starting point for this exploration is the following game-theoretic result. Lemma 3.1 ([2]). Let X be a separable metric space and let n be a nonnegative integer. The following are equivalent: (1) dim(x) = n. (2) tp Sc(O,O)(X) = n + 1.

15 112 L. BABINKOSTOVA We define for separable metric space X the game dimension of X, denoted dim G (X), by 1 + dim G (X) = tp Sc(O,O)(X). Then for X a finite-dimensional separable metric space, dim G (X) = dim(x). By a theorem of Keiô Nagami [11] and Yu. M. Smirnov [14] (independently), each separable metric space is the union of ω 1 zero-dimensional subsets. This implies the following theorem. Theorem 3.2. For each separable metric space X, dim G (X) ω 1. The following three results are easy to prove. Theorem 3.3. If X is a separable metric space with dim G (X) < ω 1, then X has property S c (O, O). Since the Hilbert cube H = [0, 1] N does not have the property S c (O, O), we have dim G (H) = ω 1. Lemma 3.4 (Subspace Lemma). Let Y be a closed subspace of X. Then dim G (Y ) dim G (X). Lemma 3.5 (Addition Lemma). Let Y X with dim G (Y ) = β, and α is minimal such that, for each closed set C X Y, we have dim G (C) α. Then dim G (X) = β + α. We have the following particularly satisfying property of game dimension in the case of countable dimensional spaces. Lemma 3.6 ([2]). Let X be a separable metric space. The following are equivalent: (1) X is countable dimensional. (2) dim G (X) = ω. Now consider the behavior of game dimension on spaces X with ω < dim G (X) < ω 1. As noted in Theorem 3.3, these spaces are among the spaces with property S c (O, O). A number of important subclasses of the spaces with property S c (O, O) have been identified and play an important role in developing game dimension. The following results point out some of these connections. Theorem 3.7. S c (O, O) = S c (O kfd, O) = S c (O fd, O).

16 TOPOLOGICAL GAMES AND COVERING DIMENSION 113 Proof: We show that S c (O fd, O) S c (O, O). Let X satisfy S c (O fd, O). Let (U n : n < ) be a sequence of open covers of X. Now we use the technique in [3] to convert this sequence of open covers to a sequence of FD-covers: Write N = k< Y k where each Y k is infinite and the Y k s are pairwise disjoint. Fix a k, a finite dimensional subset C of the space, and a set I Y k with I = dim(c) + 1. Then for each i I, U i is an open cover of C and we can find a pairwise disjoint refinement V i of U i, where V i consists of sets open in the relative topology of C, such that i I V i is an open cover of C. Without loss of generality, we may assume that the sets in V i are open in X and pairwise disjoint in X. Define U(C, I) = ( i I V i ). Finally, set V k = {U(C, I) : C X finite dimensional and I Y k with I = dim(c) + 1}. Then each V k is in O fd. Applying S c (O fd, O) to the sequence (V k : k < ), we find a sequence (H k : k < ) so that each H k is pairwise disjoint, refines V k, and k< H k is an open cover of X. For each k, and for each W H k, choose a finite dimensional set C W and a finite set I W Y k so that W U(C W, I W ). For each i I W, choose V i (W ) an open pairwise disjoint refinement of U i so that U(C W, I W ) = ( i I W V i (W )). Put I k = W H k I W, a subset of Y k. For each i I k, define G i = {W V : W H i and V V i (W )}. Each G i is a pairwise disjoint open refinement of the corresponding U i, and the union of all the H i s is an open cover of X. The inequalities in the following theorem can be proved also for the family O kfd substituting for O fd. However, the result given here seems optimal. Theorem 3.8. Let X be an infinite dimensional metrizable space. Then { tps1 (O dim G (X) fd,o)(x) tp S1 (O fd,o)(x) a limit ordinal tp S1 (O fd,o)(x) + ω otherwise. Proof: Let α be an ordinal such that TWO has a winning strategy in the game G α 1 (O fd, O). Let F be a winning strategy for TWO. We

17 114 L. BABINKOSTOVA shall use F to define a winning strategy for TWO in G β c (O, O) for the appropriate β. We shall repeatedly use Lemma 1.2 and Lemma 3.1. We may assume without loss of generality that the open covers played by player ONE of G β c (O, O) never include X as an element. To begin, choose, by Lemma 1.2, a finite dimensional set C 0 X such that there is for each proper open set U C 0 an O O fd with F (O) = U. Put n 0 = dim(c 0 ) + 1. During the first n 0 innings of G β c (O, O), TWO chooses for each O j O a disjoint open refinement T j such that C 0 W 0 = ( j n 0 T j ) (this uses Lemma 3.1). Since X is infinite dimensional, X = W 0. Then choose a B 0 O fd with W 0 = F (B 0 ). For the second stage of the game, choose, by Lemma 1.2, a finite dimensional set C 1 X such that there is for each proper open set U C 1 a U O fd with U = F (B 1, U). Put n 1 = dim(c 1 ) + 1. During the next n 1 innings of G β c (O, O), TWO chooses for each O j O a disjoint open refinement T j of O j such that C 1 W 1 = ( n0 +n 1 j=n 0 +1 T j) (this uses Lemma 3.1). As before, X = W 1. Choose B 1 O fd with F (B 0, B 1 ) = W 1. Continuing like this, we find that after ω innings, we have chosen finite dimensional subsets C 0,, C j, of X, open disjoint refinements T j of open covers O j, and nonnegative integers n 0,, n j,, j < ω, and elements B j of O fd such that for all j 0, (1) n j = dim(c j ) + 1; (2) C j ( n j 1 +n j i=n j 1 +1) T i) = W j = X; (3) F (B 0,, B j ) = W j. Assume that ν < α is an infinite ordinal, and that for each ρ < ν, we have selected a finite dimensional set C ρ, an element B ρ of O fd, an element W ρ of B ρ, an open cover O ρ, and a disjoint open refinement T ρ of O ρ such that the following hold. Write ρ as ρ = ω β 1 n ω β k n k where now β 1 > > β k 0 and each n i is positive. (1) C ρ is such that for each proper open set U C ρ there is a B O fd with U = F ((B σ : σ < ρ) (B)); (2) If β k = 0, put μ = sup{σ < ρ : σ a limit ordinal}, so that ρ = μ + n k. Set m = μ j<μ+n k (dim(c j ) + 1). Then we already have available O μ,, O μ+m+dim(cρ )+1, and

18 TOPOLOGICAL GAMES AND COVERING DIMENSION 115 T μ,, T μ+m+dim(cρ )+1 with S ρ ( μ+m+dim(c ρ)+1 i=μ+m+1 T i ) = W ρ ; (3) If β k > 0, then ρ is a limit ordinal. Set m = dim(c ρ ) + 1. Then we already have available O ρ+1,, O ρ+m and T ρ+1,, T ρ+m with S ρ ( ρ+m i=ρ+1 T i) = W ρ. (4) W ρ = F (B σ : σ ρ). We now describe what happens at ν. Case 1: ν is a limit ordinal. By Lemma 1.2, choose a finite dimensional subset C ν of X such that for each proper open set U C ν there is a U O fd with U = F ((B ρ : ρ < ν) (U)) and put m = dim(c ν ). In innings j {ν, ν + 1,, ν + m} of G β c (O, O), choose open disjoint refinements T j of O j such that C ν ( ν+m j=ν T j) = W ν (this uses Lemma 3.1). Since X is infinite dimensional, W ν = X, so choose by Lemma 1.2 a B ν O fd with W ν = F (B ρ : ρ ν). Case 2: ν is a successor ordinal. Write ν = ρ + k + 1 for some nonnegative integer k and limit ordinal ρ. By the induction hypothesis, we have finite dimensional sets C ρ,, C ρ+k available, and with l = ρ+k j=ρ (dim(c j) + 1), we also have available the disjoint open refinements T ρ,, T ρ+l of the open covers O ρ,, O ρ+l of X. Now choose, by Lemma 1.2, a finite dimensional subset C ν of X such that for each proper open set U C ν there is a U O fd such that U = F ((B σ : σ < ν) (U)). Put m = dim(c ν ) + 1. During the next m innings of G β c (O, O), TWO chooses open disjoint refinements T ρ+l+i of O ρ+l+i, 1 i m, so that C ν ( ρ+l+m i=ρ+l+1 T i) = W ν = X (this uses Lemma 3.1), and then chooses B ν O fd such that W ν = F (B σ : σ ν). Thus, we see that the recursive conditions hold also at ν. Thus, for a play ((O σ, T σ ) : σ < β) of G β c (O, O) played according to this strategy, we find a sequence ((B ν, W ν ) : ν < α) which is an F -play of G α 1 (O fd, O), thus won by TWO, for which the union of the set of W ν is covered by the union of the set of T σ s. Moreover, the strategy described shows that if α is a limit ordinal, then β = α works, and if α is a successor ordinal, then TWO wins G β c (O, O) at some inning before α + ω. Since tp S1 (O fd,o)(x) tp S1 (O kfd,o)(x), we find the following.

19 116 L. BABINKOSTOVA Corollary 3.9. For X an infinite dimensional separable metric space, { tps1 (O dim G (X) kfd,o)(x) tp S1 (O kfd,o)(x) a limit ordinal tp S1 (O kfd,o)(x) + ω otherwise. We now demonstrate a few features of these results. Using Lemma 3.6 for the Smirnov compactum S ω, we have that tp S1 (O kfd,o)(s ω ) = 2 < dim G (S ω ) = ω. Thus, the second alternative of Corollary 3.9 cannot be improved. Since K is not countable dimensional, Lemma 3.6 implies that dim G (K) ω + 1. It is also easy to see that dim G (K) ω + 1, and thus dim G (K) = ω + 1. In the first ω innings TWO covers L. The remaining uncovered piece of K is a compact subset of the totally disconnected part M of K, and thus is zero-dimensional (which TWO covers in one more inning). We also have tp S1 (O kfd,o)(k) = ω + 1. But the second alternative of Corollary 3.9 implies that dim G (K) (ω + 1) + ω = ω 2. Thus, the second alternative of Corollary 3.9 does not give optimal information in this case. Proposition Let X and Y be compact metrizable spaces. For each open cover Uof X Y, there are open covers A of X and B of Y such that {A B : A A and B B} refines U. Proof: Let d X be a compatible metric on X and let d Y be a compatible metric on Y. Let d be a metric on X Y defined as d((x 1, y 1 ), (x 2, y 2 )) = d X (x 1, x 2 ) 2 + d Y (y 1, y 2 ) 2. Then d is a compatible metric on X Y. Consider an open cover U of X Y. Since X Y is compact, choose by the Lebesgue Covering Lemma a positive real number δ such that, for each subset S X Y with d-diameter less than δ, there is a U U with S U. Let V be a finite open cover of X consisting of sets of d X -diameter less than δ 2, and let B be a finite open cover of Y consisting of open sets of d Y -diameter less than δ 2. Then {A B : A A and B B} is an open cover of X Y and consists of sets of d-diameter less than δ. Thus, this open cover refines U. Proposition For each n, dim G (K I n ) ω + n + 1.

20 TOPOLOGICAL GAMES AND COVERING DIMENSION 117 Proof: Write I n = A 0 A 1 A n where each A i is zerodimensional. By Proposition 3.10, we may assume the open covers U of K I n played by ONE are finite, of the form {A B : A A and B B} where A is an open cover of K and B is an open cover of I n. Now consider the following strategy for TWO: Let F be TWO s winning strategy in G ω+1 c (O, O) on K. In even indexed innings (including index 0), when TWO considers the moves (O 0,, O 2 n ) of ONE, TWO plays TK 2 n = F (OK 0, O2 K,, O2 n K ) on K. Since A 0 I is zero-dimensional, TWO chooses an open refinement TI 2 n of OI 2 n which covers A 0. TWO plays T 2 n = G(O 0, O 1,, O 2 n 1, O 2 n ) = {U V : U TK 2 n 2 n, V TI }. In odd innings, the same plan is used on A 1 instead of A 0. Consider the status after ω innings have elapsed. The uncovered part is contained in a set of the form C I n where C is compact and zero-dimensional. Thus, it takes at most n + 1 more innings. This shows that dim G (K I n ) ω + n + 1. Continuing investigating game dimension s product theory, we also note the following. Theorem If X is a compact metrizable space and C is the Cantor set, then dim G (X) = dim G (X C). Proof: When U is an open cover of X C, we may assume by Proposition 3.10 that U is finite and that there are finite open covers U X of X and U C of the Cantor set such that U = {U V : U U X and V U C }. Since C is zero dimensional, we may assume that U C is pairwise disjoint. Let α denote dim G (X). We may assume that α > ω. Let F be TWO s winning strategy in G α c (O, O) on X. We now define a winning strategy G for TWO in the game G α c (O, O) played on X C. With γ < α and open covers (U 0,, U γ ) of X C given, each finite, compute V γ X = F (U X 0,, U γ X ), and compute U γ C, and then define G(U 0,, U γ ) = {U V : U V γ X and V U γ C }. This defines G for inning γ for each γ < α. To see that G is a winning strategy, consider a G-play of length α, ONE s moves denoted U γ, γ < α, and TWO s, T γ.

21 118 L. BABINKOSTOVA Consider an (x, c) X C. From the definition of G, we have an F -play U γ X, T γ X = F (U δ X : δ γ), γ < α on X, won by TWO. Choose γ < α with x T γ X. Pick a V T γ X with x V. Also, choose a C U γ C with c C. Then V C is in T γ. Thus, G is a winning strategy for TWO, showing that dim G (X C) α. Since X is homeomorphic to a closed subspace of X C, Lemma 3.4 implies that α = dim G (X) dim G (X C). Theorem Let X and Y be compact metrizable spaces. If dim G (X) is a limit ordinal and Y is countable dimensional, then dim G (X) = dim G (X Y ) Proof: Put α = dim G (X). Write α = n< S n where each S n has order type α, and for m < n, S m S n =. For each n, enumerate S n in an order preserving way as (s n γ : γ < α). Write Y = n< Y n where each Y n is a zero-dimensional set. Let F be a winning strategy for TWO in the game G α c (O, O) on X. We define a winning strategy G for Y in the game G α c (O, O) on X Y : First, when ONE plays an open cover U of X Y, TWO will replace it, using Proposition 3.10, with a finite refinement of the form {U V : U U X and V U Y } where U X is an open cover of X and U Y is an open cover of Y. Now we define G. Let an inning γ < α be given, as well as a move O γ of ONE. Choose n with γ S n, say γ = s n ν. Replace O γ with the refinement {U V : U O γ X and V Oγ Y } where Oγ X is a finite open cover of X and O γ Y is a finite open cover of Y. Let V γ be a finite disjoint refinement of O γ Y which covers Y n, and let T γ X = F (Oγ X,s n,, O γ 0 X,s ). Then define n ν T γ = G(O ν : ν γ) := {U V : U T γ X and V V γ}. It follows that G is a winning strategy for TWO in G α c (O, O), showing dim G (X Y ) α. But X is homeomorphic to a closed subspace of X Y, implying by Lemma 3.4 that α dim G (X Y ). As a corollary of Theorem 3.13, we have the following. Corollary If X is a compact metric space with dim G (X) a countable limit ordinal, then dim G (X K) dim G (X) 2.

22 TOPOLOGICAL GAMES AND COVERING DIMENSION 119 The product theory for game dimension is not well behaved for complete metric spaces. In [10], Jan van Mill and Pol construct a complete metric space X such that X satisfies S c (O, O), but X X is strongly infinite dimensional. Examining their proof of their Theorem 3.1 that X has property S c (O, O), we see that, in fact, dim G (X) ω 2. Thus, X is a complete metric space with dim G (X) ω 2, while dim G (X X) = ω Final remarks In this paper we found mostly upper bounds on game length or game dimension. We suspect that some of those upper bounds are actually also the lower bounds. We have the following questions. In connection with Theorem 2.6, we ask, Problem 4.1. Is it true that for each n, tp S1 (O kfd,o)(k n ) = ω n+1? The case m = 1 of Lemma 2.8 shows that the value of tp S1 (O kfd,o)(x Y ) obtained in Theorem 2.9 cannot be improved. We suspect that the upper bound in Theorem 2.9 is, in fact, achieved. Problem 4.2. If X is a metrizable space with tp S1 (O kfd,o)(x) an infinite successor ordinal and Y is a metric space with tp S1 (O kfd,o)(y ) ω, does it follow that tp S1 (O kfd,o)(x Y ) = tp S1 (O kfd,o)(x) + ω? In connection with Proposition 3.11, we ask, Problem 4.3. Is it true that for each n, tp S1 (O kfd,o)(k I n ) = ω + n + 1? By Theorem 2.6 and Corollary 3.9, we have dim G (K n ) ω (n + 1). We suspect that the actual game dimension is smaller and ask, Problem 4.4. Is it true that for each n, dim G (K n ) = ω n + 1? References [1] David F. Addis and John H. Gresham, A class of infinite-dimensional spaces. I. Dimension theory and Alexandroff s problem, Fund. Math. 101 (1978), no. 3, [2] Liljana Babinkostova, Selective screenability game and covering dimension, Topology Proc. 29 (2005), no. 1,

23 120 L. BABINKOSTOVA [3], When does the Haver property imply selective screenability? Topology Appl. 154 (2007), no. 9, [4] Liljana Babinkostova and Marion Scheepers, Selection principles and countable dimension, Note Mat. 27 (2007), suppl. 1, [5] Piet Borst, Some remarks concerning C-spaces, Topology Appl. 154 (2007), no. 3, [6] Stewart Baldwin, Possible point-open types of subsets of the reals, Topology Appl. 38 (1991), no. 3, [7] Andrew J. Berner and István Juhász, Point-picking games and HFDs in Models and Sets (Aachen, 1983). Ed. G. H. Müller and M. M. Richter. Lecture Notes in Mathematics, Berlin: Springer, [8] Peg Daniels and Gary Gruenhage, The point-open type of subsets of the reals, Topology Appl. 37 (1990), no. 1, [9] Ryszard Engelking and Elżbieta Pol, Countable-dimensional spaces: a survey, Dissertationes Math. (Rozprawy Mat.) 216 (1983). [10] Jan van Mill and Roman Pol, A complete C-space whose square is strongly infinite-dimensional, Israel J. Math. 154 (2006), [11] Keiô Nagami, Monotone sequence of 0-dimensional subsets of metric spaces, Proc. Japan Acad. 41 (1965), [12] Roman Pol, A weakly infinite-dimensional compactum which is not countable-dimensional, Proc. Amer. Math. Soc. 82 (1981), no. 4, [13], On classification of weakly infinite-dimensional compacta, Fund. Math. 116 (1983), no. 3, [14] Yu. M. Smirnov, On universal spaces for certain classes of infinite dimensional spaces, Amer. Math. Soc. Transl. (2) 21 (1962), [15] Marion Scheepers, A topological space could have infinite successor pointopen type, Topology Appl. 61 (1995), no. 1, [16], The length of some diagonalization games, Arch. Math. Logic 38 (1999), no. 2, Department of Mathematics; Boise State University; Boise, ID address: liljanababinkostova@boisestate.edu