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Given a topological space $\mathcal{X}=(X,\tau)$, the Banach-Mazur game on $\mathcal{X}$ is the (two-player, perfect information, length-$\omega$) game played as follows: Players $1$ and $2$ alternately … My question is: Does ZF alone prove that there is some space $\mathcal{X}$ whose Banach-Mazur game is undetermined? …

The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. … The Banach-Mazur game on $\mathbb P$ is $\omega$-strategically closed when there is a strategy $\sigma$ such that II wins whenever II plays according to $\sigma$, meaning if the sequence of plays so far …

The Banach-Mazur game, also known as the $**$-game shows (and is equivalent to) that every set of reals has the Baire property. … and closed under Borel ($\Delta^1_1$) substitutions then we have $$Det(\Gamma) \rightarrow\forall A\in \Gamma, G^{**}(A)\text{ is determined}$$ This can be shown by noting that the payoff set of the Banach-Mazur …

The Wikipedia article on the Axiom of Determinacy (AD) claims: Equivalent to the axiom of determinacy is the statement that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is … In the Banach-Mazur game $BM(X)$ with payoff set $X\subseteq \mathbb{R}$, the players build a descending sequence $U_0\supseteq U_1\supseteq U_2\supseteq \dots$ by alternately choosing nonempty open sets …

The answer to the topological version of Banach-Mazur games is negative, proved by Gabriel Debs in 1985: Debs, Gabriel, Stratégies gagnantes dans certains jeux topologiques (Winning strategies in certain … www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/126/1/104687/strategies-gagnantes-dans-certains-jeux-topologiques The paper is in French, but here is the MathScinet review: A Banach-Mazur …

ZF + DC + 'every Banach-Mazur game is determined' is inconsistent. … Therefore ZF + DC proves that there is an undetermined Banach-Mazur game. Right now I don't see how to use a failure of DC to build an undetermined game. …

So we can separately ask: Version 2: Does ZF prove "There is an un-quasidetermined Banach-Mazur game?" … Version 3: Is the statement "Every Banach-Mazur game is weakly determined" consistent with ZF? …

The Banach-Mazur game is an example of a game in a topological setting. There are various other games of this nature which are mostly related to foundational questions in point-set topology. … I'm afraid I can't think of a meaningful connection between game theory and algebraic topology. …

Banach as Problem 67 in The Scottish Book, which is available here. The following text is copied from R. Daniel Mauldin's edition: (A MODIFICATION OF MAZUR'S game, [see Problem 43]). … (2) The game, similar to the one above, with the assumption that $E_i=1/2[E_0-E_1-\dots-E_{i-1}]\ i=2,3,\dots$ ad inf., and $E_1=(1/2)E_0.$ Player $A$ wins if $E_1+E_2+\dots=E_0.$ Is there a method …

and Choquet game, similarly as in [K, Chapter 8]. … Some basic fact about $\alpha$-favorable spaces are also mentioned in the article The Banach-Mazur Game at Dan Ma's topology blog, although the name "Choquet game" is not used there. [D] J. …

Games appear in pure mathematics, for example, Ehrenfeucht–Fraïssé game (in mathematical logic) and Banach–Mazur game (in topology). … Are there applications of advanced (anything behind the basic definitions) game theory ideas in pure mathematics? Thanks! …

player I's move in that game, compute the response $r$ of $\sigma$ to that play, and then extend $r$ to a chosen condition $r^+$ whose label codes the sequence $p^{\to},q$. … $\Box$ This answer amounts to the partial-order analogue of the Debs result I mentioned in my other answer, which he proved for the topological Banach-Mazur games. …

Intuitively, a sufficiently $\mathbb{P}$-generic filter yields a sufficiently good stationary Banach-Mazur strategy for player $2$ (and both stationarity and player-$2$-winning are appropriately WLOG here … There is some set $A\subseteq\omega^\omega$ whose Banach-Mazur game $G_{BM}(A)$ is determined such that any winning strategy for $G_{BM}(A)$ computes $\alpha$. …

The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I and II play an inning per positive integer. … Some facts about the Banach-Mazur game: A nonempty topological space $X$ is a Baire space if and only if Player I has no winning strategy in the Banach-Mazur game $\textsf{BM}(X)$. …

Suppose Player II has a winning strategy $\sigma$ in the Banach–Mazur game on a poset $P$. …