In mathematics, a topological space ${\displaystyle X}$ is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, and analysis, in particular functional analysis. For more motivation and applications, see the article Baire category theorem. The current article focuses more on characterizations and basic properties of Baire spaces per se.

Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ in his 1899 thesis.

The definition that follows is based on the notions of a meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior, i.e., nowhere dense sets) and a nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details.

A topological space ${\displaystyle X}$ is called a Baire space if it satisfies any of the following equivalent conditions:

The equivalence between these definitions is based on the associated properties of complementary subsets of ${\displaystyle X}$ (that is, of a set ${\displaystyle A\subseteq X}$ and of its complement ${\displaystyle X\setminus A}$) as given in the table below.

The Baire space is kind of the qualitative version of the measure space. For example, the definition 6 above is analogous to the following fact for measure spaces: Whenever a countable union of sets has positive measure, at least one of the sets has positive measure. The advantage of the Baire category approach is that it works well in infinite dimensional cases, where the measure-theoretic approach runs into significant difficulties. The table below shows more ideas they share. However, they are not mathematically equivalent. There exist meagre sets that have positive Lebesgue measure.

The Baire category theorem gives sufficient conditions for a topological space to be a Baire space.

BCT1 shows that the following are Baire spaces: