= Baire category theorem
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Every complete metric space is a Baire space: every countable intersection of open dense subsets is dense. Equivalently, no nonempty open subset is a countable union of nowhere-dense subsets.
To prove the dense-intersection form, let $V$ be nonempty and open and let $G_1,G_2,\ldots$ be open dense sets. Successively choose closed balls
$$
\overline B(x_n,r_n)\subset B(x_{n-1},r_{n-1})\cap G_n,
\qquad 0<r_n<2^{-n},
$$
starting with a ball contained in $V\cap G_1$. The centres form a Cauchy sequence. Its limit belongs to every closed ball, and hence to $V\cap\bigcap_nG_n$.
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