Here is an outline of the talk I gave at the AU set-theoretic topology seminar on 2017 Feb 06.

- Definition of the Banach-Mazur Game:
- Introduced by Mazur as Problem 43 of the Scottish Book [pdf], a notebook kept by mathematicians frequenting the Scottish Cafe in Lwów, Poland in the 1930s and 1940s.
- Let \(S\subseteq\mathbb R\). Two players alternate choosing decreasing intervals \(N_0\supseteq E_0\supseteq N_1\supseteq E_1\supseteq\dots\). The first player wins if \(\bigcap_{n<\omega}N_n\) contains a point of \(S\); the first player wins otherwise.
- Banach (1935) showed that the second player has a winning strategy for
this game if and only if \(S\) is
*meager*, a countable union of nowhere-dense sets. The prize was a bottle of wine, awarded by Mazur. - Our variant of interest: two players choose

decreasing non-empty open subsets \(E_0\supseteq N_0\supseteq E_1\supseteq N_1\supseteq\dots\) of a topological space \(X\), and let the*second*player win if and only if \(\bigcap_{n<\omega}N_n\not=\emptyset\). Call this game \(BM(X)\).

- Limited information strategies
- Let \(M\) be a set of possible moves for a game.
- A (perfect information) strategy is a function \(\sigma:M^{<\omega}\to M\) that determines the next move for a player based upon all the previous moves of her opponent.
- A \(k\)-tactical strategy is a function \(\sigma:M^{\leq k}\to M\) that determines the next move for a player based upon the most recent \(k\) previous moves of her opponent.
- A \(k\)-Markov strategy is a function \(\sigma:M^{\leq k}\times\omega\to M\) that determines the next move for a player based upon the most recent \(k\) moves of her opponent and the current round number.
- A coding strategy is a function \(\sigma:M^{\leq 2}\to M\) that determines the next move for a player based upon the most recent moves of both players.
- A strategy is
*winning*if a player that follows the strategy is guaranteed to win the game regardless of the moves of the opponent. - A winning (perfect information) strategy for the second player in \(BM(X)\) may always be improved to a winning coding strategy. (Debs 1985; Galvin,Telgarsky 1986)
- A winning Markov (that is, \(1\)-Markov) strategy for the second player in \(BM(X)\) may always be improved to a winning tactical (that is, \(1\)-tactical, also known as stationary) strategy. (Galvin,Telgarsky 1986)
- Debs (1985) found examples of spaces for which the second player in \(BM(X)\) has winning (\(2\)-tactical; see Bartoszynski,Scheepers,Just 1993) strategies, but no winning tactical strategies.
- Telgarksy conjectures (1987) that there exist spaces \(X_k\) for \(k<\omega\) such that the second player in \(BM(X_k)\) has a winning \((k+1)\)-tactical strategy, but no winning \(k\)-tactical strategy.

- Countable/Finite Games
- Scheepers (1992) published the first of many papers under the
title of
*Meager-Nowhere Dense Games: \(n\)-Tactics*. - Let \(MG(X)\) be a game where the first player chooses meager subsets \(M_{n+1}\supseteq M_n\) during round \(n+1\), followed by the second player choosing a nowhere dense subset \(N_{n+1}\). The second player wins if \(\bigcup_{n<\omega}N_n\supseteq\bigcup_{n<\omega}M_n\).
- The special case where \(X=\kappa\) has the co-finite topology
(and therefore the first player chooses countable sets and the second
player chooses finite sets) was studied by Scheepers in
*Concerning \(n\)-tactics in the countable-finite game*. Call this game \(Sch^{\cup,\supsetneq}(\kappa)\). - Scheepers showed that any \((k+3)\)-tactical strategy for the second player in \(Sch^{\cup,\supsetneq}(\kappa)\) may be improved to a \(k\)-tactical strategy.
- Modification: \(Sch^{\cap}(\kappa)\): the first player chooses any countable set \(C_n\), and the second player chooses any finite set \(F_n\). The second player wins if \(\bigcup_{n<\omega}F_n\supseteq\bigcap_{n<\omega}C_n\).
- The axiom \(\mathcal A’(\kappa)\) implies that the second player has a winning \(2\)-tactical strategy in \(Sch^{\cup,\supsetneq}(\kappa)\) (Scheepers) and a winning \(2\)-Markov strategy in \(Sch^{\cap}(\kappa)\) (Clontz).
- \(A’(\aleph_n)\) holds in ZFC for \(n<\omega\) (Clontz,Dow to appear); \(A’(\kappa)\) may be forced to hold for arbitrarily large \(\kappa\leq\mathfrak c\) (Scheepers).

- Scheepers (1992) published the first of many papers under the
title of
- Selection Games
- Let \(G_{fin}(\mathcal A,\mathcal B)\) be the game where the first player chooses \(A_n\in \mathcal A\) and the second player chooses a finite subset \(B_n\in[A_n]^{<\omega}\). The second player wins if \(\bigcup_{n<\omega}B_n\in\mathcal B\).
- For each cardinal \(\kappa\), let \(L(\kappa)=\kappa\cup\{\infty\}\) be the space with \(\kappa\) discrete and \(\infty\) having co-countable neighborhoods.
- Let the Menger game be \(Men(X)=G_{fin}(\mathcal O,\mathcal O)\) where \(\mathcal O\) is the set of open covers of \(X\).
- In regards to \(k\)-Markov strategies for the second player, \(Men(L(\kappa))\) and \(Sch^{\cap}(\kappa)\) are equivalent (Clontz).

- Markov strategies in Selection Games
- Any \((k+2)\)-Markov strategy for the second player in \(G_{fin}(\mathcal A,\mathcal B)\) may be improved to a \(2\)-Markov strategy.
- Assume \(|\bigcup\mathcal A|\leq\aleph_0\) and \(\mathcal B\) is closed under supersets. Then any winning (perfect-information) strategy for the second player in \(G_{fin}(\mathcal A,\mathcal B)\) may be improved to a Markov strategy.
- Corollaries:
- Let \(\mathcal D\) give the dense subsets of a space \(X\). The second player having a winning strategy in \(G_{fin}(\mathcal D,\mathcal D)\) characterizes strategic selective separability, \(SS^+\). Thus all countable \(SS^+\) spaces are Markov selectively separable, \(SS^{+mark}\). (Barman,Dow 2012).
- The second player having a winning strategy in \(G_{fin}(\mathcal O,\mathcal O)\) characterizes the strategic Menger property. All second-countable strategic Menger spaces are Markov Menger. Markov Menger spaces are exactly the \(\sigma\)-relatively-compact spaces (equivalent to \(\sigma\)-compact for regular spaces). (Clontz)

- Question (Gruenhage): Does there exist an \(SS^+\) space that is not \(SS^{+mark}\)?