Property of Baire

A subset ${\displaystyle A\subseteq X}$ of a topological space ${\displaystyle X}$ is called almost open and is said to have the property of Baire or the Baire property if there is an open set ${\displaystyle U\subseteq X}$ such that ${\displaystyle A\bigtriangleup U}$ is a meager subset, where ${\displaystyle \bigtriangleup }$ denotes the symmetric difference.[1] Further, ${\displaystyle A}$ has the Baire property in the restricted sense if for every subset ${\displaystyle E}$ of ${\displaystyle X}$ the intersection ${\displaystyle A\cap E}$ has the Baire property relative to ${\displaystyle E}$.[2]

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 ${\displaystyle \Gamma }$ is determined, then every set in ${\displaystyle \Gamma }$ 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.[3]

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.[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.[5]

• Almost open map – Map that satisfies a condition similar to that of being an open map.
• Baire category theorem – On topological spaces where the intersection of countably many dense open sets is dense
• Open set – Basic subset of a topological space

• Springer Encyclopaedia of Mathematics article on Baire property