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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 BanachMazur 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 \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, \cdots of W such that W_0 \supset W_1 \supset \cdots. The player P_1 wins if and only if X \cap (\cap_ W_n) \neq \emptyset.
The following properties hold.
* P_2 \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 \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, \cdots) if and only if \cap_ W_n \neq \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 \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 \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, \cdots of J such that J_0 \supset J_1 \supset \cdots, and P_1 wins if and only if X \cap (\cap_ J_n) \neq \emptyset. P_2 wins if and only if X\cap (\cap_ J_n) = \emptyset.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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