Baire spaceIn topology, a Baire space is a topological space in which, intuitively, there are "enough" points for certain limit processes.
A topological space X is called a Baire space if it satisfies one (and therefore all) of the following equivalent conditions:
- Every intersection of countably many dense open sets is dense.
- If X is non-empty, then every intersection of countably many dense open sets is also non-empty.
- The interior of every union of countably many nowhere dense sets is empty.
- Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point.
- If X is non-empty, then X is of second category. (Unions of countably many nowhere dense sets are called sets of first category or meagre; sets which are not of first category are sets of second category. Note that this notion of "category" has nothing to do with category theory.)
Examples of Baire spaces:
- all complete metric spaces are Baire spaces (this is the Baire category theorem).
- every space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space (this includes the irrational numbers)
- every locally compact Hausdorff space is a Baire space (this includes all manifolds)