# nLab Baire space

This entry is about the class of topological spaces satifiying the Baire category theorem. For the space of irrational numbers underlying Kleene's second algebra and used in computable analysis, see instead at Baire space (computability).

# Baire spaces

## Idea

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

It should not be confused with the space of irrational numbers (sometimes called ‘Baire space’ and coincidentally – or maybe not so coincidentally! i.e., not by complete accident – an example of a Baire space in our sense). 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

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

• Any locally compact Hausdorff space is a Baire space. In fact, any G-delta set of a locally compact Hausdorff space is a Baire space under the subspace topology.

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

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

• As mentioned above, the space of irrational numbers, or equivalently of infinite sequences of natural numbers, is also known as ‘Baire space’. It is a Baire space in the present sense (since it admits a complete metric), but not much should be made of the fact it has the same name. (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, so maybe it is important!)

Revised on March 1, 2014 06:17:26 by Urs Schreiber (89.204.137.245)