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Axiom of determinacy
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==Types of games that are determined== The axiom of determinacy refers to games of the following specific form: Consider a subset ''A'' of the [[Baire space (set theory)|Baire space]] ω<sup>ω</sup> of all [[infinite sequence]]s of [[natural number]]s. Two players alternately pick natural numbers :''n''<sub>0</sub>, ''n''<sub>1</sub>, ''n''<sub>2</sub>, ''n''<sub>3</sub>, ... That generates the sequence ⟨''n''<sub>''i''</sub>⟩<sub>''i''∈ω</sub> after infinitely many moves. The player who picks first wins the game if and only if the sequence generated is an element of ''A''. The axiom of determinacy is the statement that all such games are determined. Not all games require the axiom of determinacy to prove them determined. If the set ''A'' is [[clopen]], the game is essentially a finite game, and is therefore determined. Similarly, if ''A'' is a [[closed set]], then the game is determined. By the [[Borel determinacy theorem]], games whose winning set is a [[Borel set]] are determined. It follows from the existence of sufficiently [[large cardinal]]s that AD holds in [[L(R)]] and that a game is determined if it has a [[projective set]] as its winning set (see [[Projective determinacy]]). The axiom of determinacy implies that for every subspace ''X'' of the [[Real line#As a topological space|real numbers]], the [[Banach–Mazur game]] BM(''X'') is determined, and consequently, that every set of reals has the [[property of Baire]].
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