Property of Baire
Property of Baire
A subset <math>A</math> of a topological space <math>X</math> 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 <math>U\subseteq X</math> such that <math>A \bigtriangleup U</math> is meager (where <math>\bigtriangleup</math> denotes the symmetric difference).
Definitions A subset <math>A \subseteq X</math> of a topological space <math>X</math> is called almost open and is said to have the property of Baire or the Baire property if there is an open set <math>U\subseteq X</math> such that <math>A \bigtriangleup U</math> is a meager subset, where <math>\bigtriangleup</math> denotes the symmetric difference.
Properties 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.
It follows from the axiom of choice that there are sets of reals without the property of Baire. In particular, a 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.