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 has been the first infinite positional game of perfect information to be studied.
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 , a fixed subset , and a family of subsets of that satisfy the following properties.
* Each member of has non-empty interior.
* Each non-empty open subset of contains a member of .
We will call this game . Two players, and , choose alternatively elements , , of such that . wins if and only if