Property of Baire

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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).[1]

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.[1] 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.[2]

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.[3] 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.[4]

See also

References

  1. 1.0 1.1 {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  2. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  3. Template:Harvtxt, p. 22.
  4. {{#invoke:citation/CS1|citation |CitationClass=citation }}. See in particular p. 64.

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