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.

Let ${\displaystyle Y}$ be topological space, ${\displaystyle X}$ a fixed subset of ${\displaystyle Y}$ and ${\displaystyle {\mathcal {W}}}$ a family of subsets of ${\displaystyle Y}$ that have the following properties:

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