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Concept
Baire space
Summary
In mathematics, a topological space is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the , 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, analysis, in particular functional analysis. For more motivation and applications, see the article . 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 in his 1899 thesis.
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 is called a Baire space if it satisfies any of the following equivalent conditions:
Every countable intersection of dense open sets is dense.
Every countable union of closed sets with empty interior has empty interior.
Every meagre set has empty interior.
Every nonempty open set is nonmeagre.
Every comeagre set is dense.
Whenever a countable union of closed sets has an interior point, at least one of the closed sets has an interior point.
The equivalence between these definitions is based on the associated properties of complementary subsets of (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) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological space is a Baire space.
(BCT2) Every locally compact regular space is a Baire space. In particular, every locally compact Hausdorff space is a Baire space.
Official source
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