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 Stanisław Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach. 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 Stanisław Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach. Let Y {displaystyle Y} be a non-empty topological space, X {displaystyle X} a fixed subset of Y {displaystyle Y} and W {displaystyle {mathcal {W}}} a family of subsets of Y {displaystyle Y} that have the following properties: Players, P 1 {displaystyle P_{1}} and P 2 {displaystyle P_{2}} alternatively choose elements from W {displaystyle {mathcal {W}}} to form a sequence W 0 ⊇ W 1 ⊇ ⋯ . {displaystyle W_{0}supseteq W_{1}supseteq cdots .} P 1 {displaystyle P_{1}} wins if and only if Otherwise, P 2 {displaystyle P_{2}} wins.This is called a general Banach–Mazur game and denoted by M B ( X , Y , W ) . {displaystyle MB(X,Y,{mathcal {W}}).}