Each non-empty open subset of contains a member of.
Players, and alternately choose elements from to form a sequence wins if and only if Otherwise, wins. This is called a general Banach–Mazur game and denoted by
Properties
has a winning strategy if and only if is of the first category in .
The siftable and strongly-siftable spaces introduced by Choquet can be defined in terms of stationary strategies in suitable modifications of the game. Let denote a modification of where is the family of all non-empty open sets in and wins a play if and only if
A Markov winning strategy for in can be reduced to a stationary winning strategy. Furthermore, if has a winning strategy in, then has a winning strategy depending only on two preceding moves. It is still an unsettled question whether a winning strategy for can be reduced to a winning strategy that depends only on the last two moves of.
is called weakly -favorable if has a winning strategy in. Then, is a Baire space if and only if has no winning strategy in. It follows that each weakly -favorable space is a Baire space.
Many other modifications and specializations of the basic game have been proposed: for a thorough account of these, refer to . The most common special case arises when and consist of all closed intervals in the unit interval. Then wins if and only if and wins if and only if. This game is denoted by
A simple proof: winning strategies
It is natural to ask for what sets does have a winning strategy in. Clearly, if is empty, has a winning strategy, therefore the question can be informally rephrased as how "small" does have to be to ensure that 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 are key to the proof: for instance, if is equipped with the discrete topology and consists of all non-empty subsets of, then has no winning strategy if . Similar effects happen if is equipped with indiscrete topology and A stronger result relates to first-order sets. This does not imply that has a winning strategy if is not meagre. In fact, has a winning strategy if and only if there is some such that is a comeagre subset of 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. Assuming the axiom of choice, there are subsets of the unit interval for which the Banach–Mazur game is not determined.
