In mathematics, a topological space 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 \R^n in his 1899 thesis.

Definition

The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details. A topological space 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 X (that is, of a set and of its complement ) as given in the table below.

Baire category theorem

The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. BCT1 shows that the following are Baire spaces: BCT2 shows that the following are Baire spaces: One should note however that there are plenty of spaces that are Baire spaces without satisfying the conditions of the Baire category theorem, as shown in the Examples section below.

Properties

Let be a sequence of continuous functions with pointwise limit If X is a Baire space, then the points where f is not continuous is in X and the set of points where f is continuous is dense in X. A special case of this is the uniform boundedness principle.

Examples

The following are examples of Baire spaces for which the Baire category theorem does not apply, because these spaces are not locally compact and not completely metrizable: Algebraic varieties with the Zariski topology are Baire spaces. An example is the affine space consisting of the set of n-tuples of complex numbers, together with the topology whose closed sets are the vanishing sets of polynomials