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.

Definition

Let Y be a non-empty topological space, X a fixed subset of Y and \mathcal{W} a family of subsets of Y that have the following properties: Players, P_1 and P_2 alternately choose elements from \mathcal{W} to form a sequence P_1 wins if and only if Otherwise, P_2 wins. This is called a general Banach–Mazur game and denoted by

Properties

Many other modifications and specializations of the basic game have been proposed: for a thorough account of these, refer to [1987]. The most common special case arises when and \mathcal{W} consist of all closed intervals in the unit interval. Then P_1 wins if and only if and P_2 wins if and only if. This game is denoted by MB(X, J).

A simple proof: winning strategies

It is natural to ask for what sets X does P_2 have a winning strategy in. Clearly, if X is empty, P_2 has a winning strategy, therefore the question can be informally rephrased as how "small" (respectively, "big") does X (respectively, the complement of X in Y) have to be to ensure that P_2 has a winning strategy. The following result gives a flavor of how the proofs used to derive the properties in the previous section work: The assumptions on Y are key to the proof: for instance, if Y={a,b,c} is equipped with the discrete topology and \mathcal{W} consists of all non-empty subsets of Y, then P_2 has no winning strategy if X={a} (as a matter of fact, her opponent has a winning strategy). Similar effects happen if Y is equipped with indiscrete topology and A stronger result relates X to first-order sets. This does not imply that P_1 has a winning strategy if X is not meagre. In fact, if Y is a complete metric space, then P_1 has a winning strategy if and only if there is some such that X \cap W_i is a comeagre subset of W_i. It may be the case that neither player has a winning strategy: let Y be the unit interval and \mathcal{W} be the family of closed intervals in the unit interval. The game is determined if the target set has the property of Baire, i.e. if it differs from an open set by a meagre set (but the converse is not true). Assuming the axiom of choice, there are subsets of the unit interval for which the Banach–Mazur game is not determined.