| Property Of Baire |
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| descriptive set theory | |
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: such that : is meager (here, Δ denotes the Symmetric Difference ). If a subset of a Polish Space has the property of Baire, then its corresponding Banach-Mazur Game is Determined . The converse does not hold; however, if every game in a given Adequate Pointclass Γ is determined, then every set in Γ has the property of Baire. Therefore it follows from Projective Determinacy , which in turn follows from sufficient Large Cardinals , that every Projective Set (in a Polish space) has the property of Baire. It follows from the Axiom Of Choice that there are sets of reals without the property of Baire. In particular, the Vitali Set does not have the property of Baire. |
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