Search Results

Player $II$ wins the run $\langle U_0, V_0, U_1, V_1, \dots \rangle$ of the Banach-Mazur game on $X$ iff $\bigcap_{n\in\omega}V_n \not = \emptyset$. … Is there a description of the class of spaces in which $II$ has a winning strategy in the Banach-Mazur game, in terms of continuous maps? …

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 $\textsf{BM}(X)$. …

strategy in $\textsf{BM} (\prod_{i\in I}X_{i}, \prod_{i\in I} \tau_{i} ).$ Where $\textsf{BM}(X)$ denotes the Banach-Mazur game played on a topological space $X$. … Remember that The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I and II play an inning per positive integer. …

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 …

But that's only because this looks like the Banach-Mazur game, which I can win if the target set is the irrationals. … This leads to the above game where $p_n = F_n \circ F_{n-1} \circ \ldots F_1(p)$. …

Some definitions right here https://dantopology.wordpress.com/2012/06/08/the-banach-mazur-game/ At first I asked the question right here but no one answered yet https://math.stackexchange.com/questions … /2531030/if-non-empty-player-has-a-winnig-strategy-in-banach-mazur-game-bmx-then-it-al Thanks for any help. …

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? …

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? …

Given $\mathcal{A}\subseteq\omega^\omega$, the Banach-Mazur game with payoff set $\mathcal{A}$ consists of players $1$ and $2$ alternately playing nonempty finite strings of naturals with player $1$ winning … On the other hand, Banach-Mazur determinacy principles are relatively weak on a set-theoretic level: "Every Banach-Mazur game is determined" adds no consistency strength to $\mathsf{ZF+DC}$. …

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! …

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 …

Some known results are the following: Bernstein sets are Baire spaces, also the Banach-Mazur game played in a Bernstein set is indeterminate. …

Kunen), then, using the Banach-Mazur game, Player I has a winning strategy in $\textsf{BM}(X^{\omega})$. … I was trying to show that Player I has a winning strategy in the game $\textsf{MB}(X^{\kappa})$, but unfortunately I still haven't got it. …

Topological spaces $X$ for which the second player (Non-empty) has a winning strategy in the Banach-Mazur game $BM(X)$ are called weakly $\alpha$-favorable by White and Choquet by Kechris. …

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. …