nLab Baire space

This entry is about the class of topological spaces satisfying the Baire category theorem. For the Baire space used in computable analysis, descriptive set theory, etc, see instead at Baire space of sequences.

Idea

A Baire space is a topological space that satisfies the conclusion of the Baire category theorem.

It should not be confused with the Baire space of sequences (which is an example of a Baire space in our sense but not a prominent one). Nor should it be confused with a Baire set (a subset somewhat analogous to a measurable set but defined by a topological property).

Definition

A Baire space is a topological space such that the intersection of any countable family of dense open subspaces is also dense. Equivalently: a space such that a countable union of closed sets each with empty interior also has empty interior.

Examples

The classical Baire category theorem states that:

• Any complete metric space (or rather its underlying topological space, hence any completely metrizable topological space) is a Baire space.

A second theorem, sometimes dubbed “BCT2” (as in Wikipedia):

• Any locally compact Hausdorff space, or indeed any locally compact sober space according to Wikipedia, is a Baire space. Moreover, any G-delta set of a locally compact Hausdorff space is a Baire space under the subspace topology.

Furthermore:

• Any open subspace of a Baire space is also a Baire space.

• Given a Baire space $X$, a dense $G_\delta$ set in $X$ (i.e. a countable intersection of dense opens) is a Baire space under the subspace topology. See Dan Ma’s blog, specifically Theorem 3 here.

• As mentioned above, the space of infinite sequences of natural numbers, or equivalently (up to topology) the space irrational numbers, is also known as ‘Baire space’. It is a Baire space in the present sense (since it admits a complete metric), but the coincidence of names appears to be just a coincidence. (It is much more important that Baire space is a Polish space than that Baire space is a Baire space. Of course, every Polish space is a Baire space too.)