# Baire category theorem

In

mathematics, the

**Baire category theorem** is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in

topology and

functional analysis.

The statement is:

- Every complete metric space is a Baire space.

The proof of the Baire category theorem uses the

axiom of choice; in fact, the Baire category theorem is

logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice.

The Baire category theorem is used in the proof of the open mapping theorem.