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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Banach-Mazur game ： ウィキペディア英語版 Banach–Mazur game In general topology, set theory and game theory, a Banach–Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spaces. This game was the first infinite positional game of perfect information to be studied. It was introduced by Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach. == Definition and properties == In what follows we will make use of the formalism defined in Topological game. A general Banach–Mazur game is defined as follows: we have a topological space $Y$, a fixed subset $X\; \backslash subset\; Y$, and a family $W$ of subsets of $Y$ that satisfy the following properties. * Each member of $W$ has non-empty interior. * Each non-empty open subset of $Y$ contains a member of $W$. We will call this game $MB(X,Y,W)$. Two players, $P\_1$ and $P\_2$, choose alternatively elements $W\_0$, $W\_1$, $\backslash cdots$ of $W$ such that $W\_0\; \backslash supset\; W\_1\; \backslash supset\; \backslash cdots$. The player $P\_1$ wins if and only if $X\; \backslash cap\; (\backslash cap\_\; W\_n)\; \backslash neq\; \backslash emptyset$. The following properties hold. * $P\_2\; \backslash uparrow\; MB(X,Y,W)$ if and only if $X$ is of the ''first category'' in $Y$ (a set is of the first category or meagre if it is the countable union of nowhere-dense sets). * Assuming that $Y$ is a complete metric space, $P\_1\; \backslash uparrow\; MB(X,Y,W)$ if and only if $X$ is comeager in some nonempty open subset of $Y$. * If $X$ has the Baire property in $Y$, then $MB(X,Y,W)$ is determined. * Any winning strategy of $P\_2$ can be reduced to a stationary winning strategy. * The siftable and strongly-siftable spaces introduced by Choquet can be defined in terms of stationary strategies in suitable modifications of the game. Let $BM(X)$ denote a modification of $MB(X,Y,W)$ where $X=Y$, $W$ is the family of all nonempty open sets in $X$, and $P\_2$ wins a play $(W\_0,\; W\_1,\; \backslash cdots)$ if and only if $\backslash cap\_\; W\_n\; \backslash neq\; \backslash emptyset$. Then $X$ is siftable if and only if $P\_2$ has a stationary winning strategy in $BM(X)$. * A Markov winning strategy for $P\_2$ in $BM(X)$ can be reduced to a stationary winning strategy. Furthermore, if $P\_2$ has a winning strategy in $BM(X)$, then $P\_2$ has a winning strategy depending only on two preceding moves. It is still an unsettled question whether a winning strategy for $P\_2$ can be reduced to a winning strategy that depends only on the last two moves of $P\_1$. * $X$ is called ''weakly $\backslash alpha$-favorable'' if $P\_2$ has a winning strategy in $BM(X)$. Then, $X$ is a Baire space if and only if $P\_1$ has no winning strategy in $BM(X)$. It follows that each weakly $\backslash alpha$-favorable space is a Baire space. Many other modifications and specializations of the basic game have been proposed: for a thorough account of these, refer to (). The most common special case, called $MB(X,J)$, consists in letting $Y\; =\; J$, i.e. the unit interval $()$, and in letting $W$ consist of all closed intervals $()$ contained in $()$. The players choose alternatively ''subintervals'' $J\_0,\; J\_1,\; \backslash cdots$ of $J$ such that $J\_0\; \backslash supset\; J\_1\; \backslash supset\; \backslash cdots$, and $P\_1$ wins if and only if $X\; \backslash cap\; (\backslash cap\_\; J\_n)\; \backslash neq\; \backslash emptyset$. $P\_2$ wins if and only if $X\backslash cap\; (\backslash cap\_\; J\_n)\; =\; \backslash emptyset$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Banach–Mazur game」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.