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.
be a non-empty topological space,
a fixed subset of
a family of subsets of
that have the following properties: Players,
alternately choose elements from
This is called a general Banach–Mazur game and denoted by
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
consist of all closed intervals in the unit interval.
This game is denoted by
It is natural to ask for what sets
has a winning strategy, therefore the question can be informally rephrased as how "small" (respectively, "big") does
The following result gives a flavor of how the proofs used to derive the properties in the previous section work: The assumptions on
are key to the proof: for instance, if
is equipped with the discrete topology and
consists of all non-empty subsets of
(as a matter of fact, her opponent has a winning strategy).
Similar effects happen if
is equipped with indiscrete topology and
A stronger result relates
is a complete metric space, then
It may be the case that neither player has a winning strategy: let
be the unit interval and
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.