BanachMazur

The Banach-Mazur game is a combinatorial game played on a topological space, typically a metric space. It was introduced by Stefan Banach and Stanisław Mazur in 1935. The game involves two players, traditionally called "Alice" and "Bob", who take turns selecting open sets within a given complete metric space.

The game proceeds as follows: Alice chooses an open set A1. Then Bob chooses an open set B1 contained within A1. Next, Alice chooses an open set A2 contained within B1, and Bob chooses an open set B2 contained within A2, and so on. Alice wins if the intersection of all chosen sets, the intersection of A1, B1, A2, B2, ..., is non-empty. Bob wins if this intersection is empty.

The Banach-Mazur game is a tool used to study the properties of topological spaces, particularly Baire spaces. A space is a Baire space if every complete metric space is a Baire space. The outcome of the Banach-Mazur game can characterize certain properties of a space. For instance, a theorem states that a topological space is a Baire space if and only if Alice has a winning strategy in the Banach-Mazur game played on that space, provided the game is played with open sets whose closures are contained within the space. This game is closely related to the concept of meager sets and the completeness of metric spaces.