Proof of Baire's Category Theorem
Every complete metric spaceis second category.
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
In this way we obtain a nested sequence of of closed sets
such that foand
Henceandexists such that
Thusis second category.