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
denotes the symmetric difference).
is called almost open and is said to have the property of Baire or the Baire property if there is an open set
has the Baire property in the restricted sense if for every subset
[2] The family of sets with the property of Baire forms a σ-algebra.
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
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.
[3] 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.
[4] 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.