Topological Games

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Topological games

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Topological Games

Examples of topological games appear explicitly in The Scottish Book. In these games there are two players, whom we will call ONE and TWO. The games are of length ω. These games never end in draws. Examples of such games appear earlier implicitly: In [7] Hurewicz introduced the following covering property for a space X: For each sequence (Un : n ∈ N) of open covers of X there is a sequence (Vn : n ∈ N)
of finite sets such that for each n we have Vn ⊆ Un , and n∈N Vn is a cover of X. This property is said to be the Menger property because for metric spaces it is equivalent to a basis property that K. Menger introduced; although such a space is also called a Hurewicz space. In [16] R. Telgarsky defined the following infinite two-person game: In the nth inning ONE first chooses an open cover On of X; then TWO responds with a finite subset Tn
⊆ On . A play (O1 , T1 , . . . , On , Tn , . . .) is won by TWO if n∈N Tn is an open cover of X; otherwise, ONE wins. Hurewicz proved that a space X has the Menger property if, and only if, ONE has no winning strategy in this game, and Telgarsky proved that a metric space X is σ -compact if, and only if, TWO has a winning strategy in this game. Sierpi´nski showed that Lusin sets of real numbers have the Menger property. Since Lusin sets are not σ -compact they are spaces where neither player has a winning strategy. The existence of a Lusin set is independent of ZFC. Fremlin and Miller show that there is a set of real numbers for which neither player has a winning strategy. This example illustrates some of the trends in the study of topological games. There are generally four basic questions to consider: (1) Does some player have a winning strategy? (2) Is some well-known topological property characterized by the existence/non-existence of a winning strategy of some player of the game under study? When the answer is yes, the game is usually a powerful tool in analysing the corresponding topological property. Many well-known topological concepts that were introduced and studied long before the games were invented, are now characterized by games. (3) In games where neither player has a winning strategy, does this situation change when the length of the game is increased? (4) In games where some player has a winning strategy, how much memory does that player really need to win the game? There are now numerous examples of topological games. Some are surveyed in [17]. We briefly describe two particularly important classes of games: nested chain games and diagonalization games. 1. Nested chain games: The Banach–Mazur game The classical Banach–Mazur game from the Scottish Book is probably the best known example of an infinite topological

game. There are several other nested chain games in the literature. In the Banach–Mazur game BM(X) on a space X, players ONE and TWO play a game which has an inning per positive integer. In the nth inning ONE chooses a nonempty open subset On of X and TWO responds by choosing a nonempty open subset Tn of On . In the (n + 1)th inning ONE chooses a nonempty open set On+1 ⊆ T n , and so on. TWO wins the play O1 , T1 , . . . , On , Tn , . . . if n∈N Tn = ∅; else, ONE wins. A space is a Baire space if any sequence of dense open subsets has dense intersection. Answering a question of Mazur, Banach proved for subspaces of the real line, and Oxtoby later for general spaces, that the space (X, τ ) is a Baire space if, and only if, ONE has no winning strategy in BM(X). Spaces for which TWO has a winning strategy in the Banach–Mazur game have the Baire property in a strong sense: if TWO has a winning strategy in BM(X), then all powers of X, endowed with the box topology, are Baire spaces. There is a rich literature of examples of spaces for which some box powers are Baire spaces, while other box powers are not Baire spaces. In the early 1970s in unpublished work F. Galvin made the following beautiful conjecture: if all powers of X, endowed with the box topology, are Baire spaces, then TWO has a winning strategy in BM(X). This conjecture in conjunction with the Banach–Oxtoby result and the result on box-products above would give a complete description of existence of winning strategies in the Banach–Mazur games in terms of Baireness of spaces. The current state of this conjecture is: If it is consistent that there is a proper class of measurable cardinals, then Galvin’s conjecture is consistent. There is no consistency result known to imply the negation of Galvin’s conjecture. There are some beautiful characterizations of spaces for which TWO has a winning strategy in BM(X). Here are two examples. In [8] Kenderov and Revalski consider for completely regular spaces X the set C ∗ (X) of bounded continuous real-valued functions on X, endowed with the topology of uniform convergence. They prove that TWO has a winning strategy in BM(X) if, and only if, the set {f ∈ C ∗ (X): f has a minimum value on X} contains a dense Gδ subset of C ∗ (X). In [9] Ma considers for completely regular locally compact spaces the set C(X) of continuous real-valued functions on X, endowed with the compact-open topology. Let Ck (X) denote this space; then TWO has a winning strategy in BM(Ck (X)) if, and only if, X is paracompact. In connection with memory requirements: In all results mentioned here the default meaning of “strategy” is “perfect

440 information strategy”. A perfect information strategy is a function which has as input the sequence of all prior moves of the opponent, and as output the response of the strategy owner. Fleissner and Kunen brought attention to the fact that when player ONE has a winning strategy in the Banach– Mazur game, then in fact player ONE has a winning strategy depending on only the most recent move of TWO. A strategy which uses as input only the most recent move of the opponent is said to be a tactic (or 1-tactic). They asked if it is also the case that when TWO has a winning strategy in BM(X), then TWO has a winning tactic. G. Debs gave counterexamples. In Debs’ counterexamples TWO does not have a winning tactic, but has a winning strategy depending on the most recent two moves of ONE. For a fixed k, call a strategy that uses as input only the most recent
k moves of the opponent a k-tactic. Telgársky conjectured that for each k  1 there is a space in which TWO does not have a winning k-tactic, but does have a winning (k + 1)-tactic in the Banach–Mazur game. Currently there are no examples known of spaces for which TWO has a winning 3-tactic, but not a winning 2-tactic, in the Banach–Mazur game. In [5] Galvin and Telgársky study memory requirements in the Banach–Mazur game. They prove among other things that if TWO has a winning strategy in the Banach–Mazur game, then TWO has a winning strategy which uses as information only the most recent move of ONE and the most recent move of TWO.

2. Diagonalization games The class of diagonalization games is at least as important as the nested chain games and also has a long history. An unusually large number of games in the literature can be reformulated as diagonalization games. Let A and B be families of subsets of an infinite set S. The symbol Gfin (A, B) denotes the game which has an inning per positive integer, and in the nth inning ONE chooses an element On of A, and TWO responds by choosing a finite set Tn ⊂
On . A play (O1 , T1 , . . . , On , Tn , . . .) is won by TWO if n∈N Tn ∈ B. One can consider Gfin (A, B) as a game-theoretic version of the selection property Sfin (A, B), which is defined as follows: For each sequence (Un : n ∈ N) of elements of A there is a sequence (V
n : n ∈ N) of finite sets such that for each n, Vn ⊂ Un , and n∈N Vn is a member of B. The symbol G1 (A, B) denotes the game which has an inning per positive integer, and in the nth inning ONE chooses an element On of A, and TWO responds by choosing an element Tn ∈ On . A play (O1 , T1 , . . . , On , Tn , . . .) is won by TWO if {Tn : n ∈ N} ∈ B. Now G1 (A, B) is a game-theoretic version of the selection property S1 (A, B), which is defined as follows: For each sequence (Un : n ∈ N) of elements of A there is a sequence (Vn : n ∈ N) such that for each n, Vn ∈ Un , and {Vn : n ∈ N} is a member of B.

Section J: Influences of other fields The existence of winning strategies for player TWO received much attention in the literature. If we let O denote the collection of open covers of a space X, then the game Gfin (O, O) is Telgarsky’s game discussed in the introduction of this article, and Sfin (O, O) is the Menger property discussed there. Thus Telgarsky’s theorem above states that for metric spaces TWO has a winning strategy in Gfin (O, O) if, and only if, the space is σ -compact. With O as above Galvin introduced the game G1 (O, O); Rothberger introduced S1 (O, O) in [12]. In [4] Galvin proved that for a firstcountable space X TWO has a winning strategy in G1 (O, O) if, and only if, X is countable. In [15] Telgársky generalized this as follows: The games he introduced there can all be reformulated as games of the form G1 (A, O) where A is a family of special open covers. For example: Let X be a T3 1 -space and let K be a collection 2 of closed proper subsets of X such that K contains all oneelement subsets of X, and is closed-hereditary. Then an open cover U of X is a K-cover if there is for each C ∈ K a U ∈ U with C ⊂ U , and X ∈ / U . Let O(K) denote the K-covers of X. The symbol DK denotes the collection of subsets of X which are representable as a union of a discrete family of sets in K. It is evident that K ⊂ DK. Here are some examples of such A considered by Telgársky and collaborators: : The ω-covers of X. An open cover U of a space is an ω-cover if X ∈ / U and there is for each finite set F ⊂ X a set U ∈ U with F ⊂ U . K: The κ-covers of X. An open cover U of a space is a κ-cover if X ∈ / U and there is for each compact proper subset C ⊂ X a set U ∈ U with C ⊂ U . ˇ : The cˇ -covers of X. An open cover U of a space is C ˇ a cˇ -cover if X ∈ / U and there is for each closed Cechcomplete proper subset C ⊂ X a set U ∈ U with C ⊂ U . D: The d-covers of non-discrete space X. An open cover U of a space is a d-cover if X ∈ / U and there is for each closed discrete proper subset C ⊂ X a set U ∈ U with C ⊂ U . Dimn : The dimn -covers of space X. An open cover U of a space is a dimn -cover if X ∈ / U and there is for each closed proper subset C ⊂ X which is normal and of covering dimension
n a set U ∈ U with C ⊂ U . Telgarsky proved that if every element of K is a Gδ -set then for completely regular X, TWO has a winning strategy in G1 (O(K), O) if, and only if, X is a union of countably many elements of K. Call a paracompact space X a Tamano space if for each paracompact space Y also X × Y is paracompact. An old question of Tamano asks to characterize the Tamano spaces. Games of the form G1 (A, O) have been useful in characterizing some subclasses of the class of Tamano spaces. In particular, let C denote the collection of compact proper subsets of a space X. If X is a paracompact space such that TWO has a winning strategy in the game G1 (O(DC), O), then X is a Tamano space (Telgarsky). The analogues of Tamano’s problem for subparacompact spaces and metacompact spaces have also been considered, and Yajima [MN, Chapter 13] proved: (1) If X is a regular

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Topological games

subparacompact space and if TWO has a winning strategy in the game G1 (O(DC), O), then for each subparacompact space Y also X × Y is subparacompact. (2) If X is a regular metacompact P -space and if TWO has a winning strategy in the game G1 (O(DC), O), then for each metacompact space Y also X × Y is metacompact. Notice that the class of Tamano spaces is closed under finite products. The same is not clear about the subclass characterized by these games. Specifically it is an open problem of Telgársky whether TWO has a winning strategy in the game G1 (O(DC), O) on X × Y if TWO has a winning strategy in this game on each of the T3 1 -spaces X and Y . 2 Some partial results are know: If X and Y are subparacompact T3 -spaces then the answer is yes. Several games that have been introduced by Gruenhage also can be reformulated as games of the form G1 (A, B). We now briefly survey some beautiful results from this research. A family M of nonempty compact subsets of a space X is said to be a moving off family if there is for each compact subset C ⊂ X a set M ∈ M such that M ∩ C = ∅. Let A denote the collection of moving off families for a space X. Let F denote the set of families G where each member of G is a nonempty compact subset of X, and G is locally finite. For a locally compact (but non-compact) Hausdorff space X, TWO has a winning strategy in G1 (A, F ) if, and only if, X is paracompact. Let a space X and a subset H of X be given and define ΩH = {S ⊂ X \ H : (for each open U ⊇ H )(S ∩ U = ∅)} and ΓH = {S ⊂ X \ H : (for each open U ⊇ H )(S \ U is finite)}. Also, recall that for a space X the symbol ∆ denotes {(x, x): x ∈ X}. Gruenhage proved the following for a compact space X and a subset H : (1) If X is countably tight then TWO has a winning strategy in G1 (ΩH , ΓH ) if, and only if, X \ H is meta-Lindelöf. (2) If X is scattered then TWO has a winning strategy in G1 (ΩH , ΓH ) if, and only if, X \ H is metacompact. (3) TWO has a winning strategy in G1 (Ω∆ , Γ∆ ) (on X2 ) if, and only if, X is Corson compact. Initially authors were occupied with the existence of winning strategies for TWO in these diagonalization games. Some nice theorems were missed by not instead considering the non-existence of winning strategies of ONE. Two fundamental results about the non-existence of winning strategies for ONE are Hurewicz’s theorem above that a space has property Sfin (O, O) if, and only if, ONE has no winning strategy in the game Gfin (O, O), and the following theorem of Pawlikowski, proved in [11]. A space X has property S1 (O, O) if, and only if, ONE has no winning strategy in G1 (O, O). The nonexistence of a winning strategy for ONE is often a Ramsey-theoretic statement. For positive integers n and k, the symbol A → (B)nk denotes the Ramseyan statement For each A ∈ A, for each function f : [A]n → {1, . . . , k}, there is a B ⊆ A such that B ∈ B, and f is constant on [B]n . Ramsey’s Theorem is this statement for the case when A = B and A is the collection of infinite subsets of the integers.

441 The symbol A → B2k denotes the statement: For each A ∈ A and for each f : [A]2 → {1, . . . , k} there
is a B ⊂ A with B ∈ B, and a partition B = n<∞ Bn of B into disjoint finite sets, and an i ∈ {1, . . . , k} such that f ({x, y}) = i whenever x and y are from different Bn ’s. A theorem of Baumgartner and Taylor states that if F is a non-principal ultrafilter on the integers, then F is a P -point if, and only if, for all k, F → F 2k . Scheepers proved: (1) ONE has no winning strategy in G1 (O, O) if, and only if, Ω → (O)22 . (2) ONE has no winning strategy in Gfin (O, O) if, and only if, Ω → O22 . There are now numerous examples of such results where ONE has no winning strategy in a game of the form Gfin (A, B) (respectively G1 (A, B)) if, and only if the corresponding selection hypothesis Sfin (A, B) (respectively S1 (A, B)) holds, if and only if, the corresponding Ramseyan statement A → B2k (respectively A → (B)nk ) holds for appropriate k and n. A large number of important topological properties have also been characterized by nonexistence of winning strategies of ONE in such games – for example, countable fan tightness in Cp (X), countable strong fan tightness, being a Lusin subset of the real line, being a strong measure zero subset of a sigma-compact metric space, and so on. In connection with the length of games, Daniels and Gruenhage define in [3] the point open type of a set of real numbers to be the least α
ω1 such that if the game G1 (O, O) is allowed to run for α innings, then TWO has a winning strategy. For example: Let L be a Lusin set of real numbers. The point-open type of L is ω + ω. To see this, note that during the first ω-innings TWO can cover a dense countable subset of L. The uncovered part of L now left is nowhere dense, and as L is a Lusin set, is countable. The remaining ω innings are used to cover these points. Following the initial results of [3], Baldwin proved in [2] that it is consistent that there is for each limit ordinal α < ω1 a set of real numbers with point-open type α. By examples presented in [13] there are topological spaces with infinite successor point-open type, but it is not known if there can be an infinite set of real numbers with successor point-open type. In this connection we have the following conjecture: The point-open type of any infinite set of reals is a limit ordinal. And games introduced by Berner and Juhász can be reformulated as games of the form G1 (A, B). For let D denote the collection of dense subsets of X. One can show that in the open-point game of Berner and Juhasz, ONE has a winning strategy, if an only if, in the game G1 (D, D), TWO has a winning strategy, and in their game TWO has a winning strategy if, and only if, in the game G1 (D, D), ONE has a winning strategy. In the case of metrizable spaces X there is then a beautiful duality theory between the game G1 (D, D) on Cp (X), and the game G1 (Ω, Ω) on X, as explored in [14]. Finally, as to memory requirements in diagonalization games: In [1] it was shown that some games of Gruenhage

442 introduced in [6] can be reformulated as games of the form G1 (A, B). Then some of Gruenhage’s results translate to statements of the form that for locally compact T2 -spaces metacompactness is equivalent to TWO having a particular type of limited memory strategy in the corresponding game, and for compact spaces Eberlein compactness is also characterized by the existence of a type of limited memory strategy of TWO in the corresponding game.

References [1] L. Babinkostova, C. Guido, L. Koˇcinac and M. Scheepers, Notes on selection principles in topology II: Metacompactness, in progress. [2] S. Baldwin, Possible point-open types of subsets of the reals, Topology Appl. 38 (1991), 219–223. [3] P. Daniels and G. Gruenhage, The point-open type of subsets of the reals, Topology Appl. 37 (1990), 53–64. [4] F. Galvin, Indeterminacy of the point-open game, Bull. Acad. Polon. Sci. 26 (1978), 445–449. [5] F. Galvin and R. Telgarsky, Stationary strategies in topological games, Topology Appl. 22 (1986), 51–69. [6] G. Gruenhage, Games, covering properties and Eberlein Compacts, Topology Appl. 23 (1986), 291–297. [7] W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1925), 401–421.

Section J: Influences of other fields [8] P.S. Kenderov and J.P. Revalski, The Banach–Mazur game and generic existence of solutions to optimization problems, Proc. Amer. Math. Soc. 118 (1993), 911– 917. [9] D.K. Ma, The Cantor tree, the γ -property, and Baire function spaces, Proc. Amer. Math. Soc. 119 (1993), 903–913. [10] R.D. Mauldin, ed., The Scottish Book: Mathematics from the Scottish Café, Birkhäuser, Boston (1981). [11] J. Pawlikowski, Undetermined sets of point-open games, Fund. Math. 144 (1994), 279–285. [12] F. Rothberger, Eine Verscharfung der Eigenschaft C, Fund. Math. 30 (1938), 50–55. [13] M. Scheepers, A topological space could have successor point-open type, Topology Appl. 61 (1995), 95–99. [14] M. Scheepers, The length of some diagonalization games, Arch. Math. Logic 38 (1999), 103–122. [15] R. Telgarsky, Spaces defined by topological games, Fund. Math. 88 (1975), 193–223. [16] R. Telgarsky, On games of Topsøe, Math. Scandinavica 54 (1984), 170–176. [17] R. Telgarsky, Topological games: On the 50th anniversary of the Banach–Mazur game, Rocky Mountain J. Math. 17 (1987), 227–276. Marion Scheepers Boise, ID, USA