Baire space

A topological space X is called a Baire space if it satisfies one (and therefore all) of the following equivalent conditions:

1. Every intersection of countably many dense open sets is dense.
2. If X is non-empty, then every intersection of countably many dense open sets is also non-empty.
3. The interior of every union of countably many nowhere dense sets is empty.
4. 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.
5. 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.)

In proofs, condition 4 is commonly used to show that certain interior points must exist.

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)

Note that the space of rational numbers with their ordinary topology are not a Baire space, since they are the union of countably many nowhere dense sets, the singletons.