Banach–Mazur game
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 formula_1 be a non-empty topological space, formula_2 a fixed subset of formula_1  and formula_4 a family of subsets of formula_1 that have the following properties: Players, formula_9 and formula_10 alternately choose elements from formula_4 to form a sequence formula_12 formula_9 wins if and only if Otherwise, formula_10 wins. This is called a general Banach–Mazur game and denoted by formula_16 Properties. 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 formula_52 and formula_4 consist of all closed intervals in the unit interval. Then formula_9 wins if and only if formula_55 and formula_10 wins  if and only if formula_57. This game is denoted by formula_58 A simple proof: winning strategies. It is natural to ask for what sets formula_2 does formula_10 have a winning strategy in formula_61. Clearly, if formula_2 is empty, formula_10 has a winning strategy, therefore the question can be informally rephrased as how "small" (respectively, "big") does formula_2 (respectively, the complement of formula_2 in formula_1) have to be to ensure that formula_10 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 formula_1 are  key to the proof: for instance, if formula_92 is equipped with the discrete topology and formula_4 consists of all non-empty subsets of formula_1, then formula_10 has no winning strategy if formula_96 (as a matter of fact, her opponent has a winning strategy). Similar effects happen if formula_1 is equipped with indiscrete topology and formula_98 A stronger result relates formula_2 to first-order sets. This does not imply that formula_9 has a winning strategy if formula_2 is not meagre. In fact, if formula_1 is a complete metric space, then formula_9 has a winning strategy if and only if there is  some formula_107 such that formula_108 is a comeagre subset of formula_109 It may be the case that neither player has a winning strategy: let formula_1 be the unit interval and formula_4 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.