# 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 Baire category theorem is used in the proof of the open mapping theorem and the uniform boundedness principle.

## Reference

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