Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Property of Baire

Difference of an open set by a meager set

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated September 30, 2025

A subset A of a topological space 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 U\subseteq X such that A\bigtriangleup U is meager (where \bigtriangleup denotes the symmetric difference).

01Definitions

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

02Properties

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. 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 \Gamma is determined, then every set in \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.

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.

Watch videos about Property of BaireExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Property of Baire”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Almost open map

In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open map. As described below, for certain broad categories of topological vector spaces, all surjective linear operators are necessarily almost open.

Baire category theorem

The Baire category theorem is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense).

Open set

In general topology and mathematical analysis, an open set is a generalization of an open interval in the real line. In a metric space, an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).