# Baire category theorem

 related topics {math, number, function}

The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

The theorem was proved by René-Louis Baire in his 1899 doctoral thesis.[1]

## Contents

### Statement of the theorem

A Baire space is a topological space with the following property: for each countable collection of open dense sets Un, their intersection ∩ Un is dense.

Note that neither of these statements implies the other, since there is a complete metric space which is not locally compact (the irrational numbers with the metric defined below), and there is a locally compact Hausdorff space which is not metrizable (uncountable Fort space). See Steen and Seebach in the references below.

• (BCT3) A non-empty complete metric space is NOT the countable union of nowhere-dense sets (i.e., sets whose closure has dense complement).

This formulation is a consequence of BCT1 and is sometimes more useful in applications. Also: if a non-empty complete metric space is the countable union of closed sets, then one of these closed sets has non empty interior.

### Relation to the axiom of choice

The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.[2]