Baire category theorem - Definition and Overview

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 and the uniform boundedness principle. It also gives a quick proof that the reals are uncountable (since the reals are a complete metric space, and hence cannot be the countable union of points).

Reference

R. Baire (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, 1--123.