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- Property_of_Baire abstract "A subset of a topological space has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set such that is meager (where Δ denotes the symmetric difference).The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set is almost open, and any countable union or intersection of almost open sets is again almost open. Since every open set is almost open (the empty set is meager), it follows that every Borel set is almost open.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.[citation needed]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. Already weaker versions of choice are sufficient: the Boolean prime ideal theorem implies that there is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, via binary representations of reals, a set of reals without the Baire property.".
- Property_of_Baire wikiPageExternalLink Baire_property.
- Property_of_Baire wikiPageID "2213768".
- Property_of_Baire wikiPageRevisionID "592255131".
- Property_of_Baire hasPhotoCollection Property_of_Baire.
- Property_of_Baire subject Category:Descriptive_set_theory.
- Property_of_Baire subject Category:Determinacy.
- Property_of_Baire comment "A subset of a topological space has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set such that is meager (where Δ denotes the symmetric difference).The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set is almost open, and any countable union or intersection of almost open sets is again almost open.".
- Property_of_Baire label "Baire-Eigenschaft".
- Property_of_Baire label "Property of Baire".
- Property_of_Baire label "Własność Baire'a".
- Property_of_Baire label "ベールの性質".
- Property_of_Baire sameAs Baire-Eigenschaft.
- Property_of_Baire sameAs ベールの性質.
- Property_of_Baire sameAs Własność_Baire'a.
- Property_of_Baire sameAs m.06wfjd.
- Property_of_Baire sameAs Q803931.
- Property_of_Baire sameAs Q803931.
- Property_of_Baire wasDerivedFrom Property_of_Baire?oldid=592255131.
- Property_of_Baire isPrimaryTopicOf Property_of_Baire.