## Proof of Baire's Category Theorem

Theorem

Every complete metric spaceis second category.

Proof

A topological spaceis said to be first category ifis the countable union of nowhere dense subsets ofAll other topological spaces are said to be second category.

Letbe first category. By definition then,where eachis a nowhere dense subset of

Sinceis nowhere dense inthere existsandsuch thatso

is open andis nowhere dense inso there existsuch thatand

Setthenand

In this way we obtain a nested sequence of of closed sets

such that fo and

Henceandexists such that

Alsofor everyand

Thusis second category.