Banach Mazur game

Definition

The Banach Mazur game (on \mathbb{R}) is a game with two players. Let A \subseteq R. The players take turns playing open sets one after another, with the condition that every open set is a subset of the set last played.

While this seems like an arbitrary game, it has interesting properties and theorems associated with it.

Theorem

For any set A \in \mathbb{R}

1. A is co-meager if and only if Player 2 has a winning strategy
2. A is meager if and only if Player 1 has a winning strategy