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Property of Baire

## Summary

A subset ${\displaystyle A}$ of a topological space ${\displaystyle X}$ 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 ${\displaystyle U\subseteq X}$ such that ${\displaystyle A{\mathbin {\Delta }}U}$ is meager (where Δ denotes the symmetric difference).[1]

## Definitions

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]

## 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.[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]