In a complete metric space or a locally compact Hausdorff space, every countable intersection of dense open sets is dense.Suggest a slogan
If \(X\) is a complete metric space or a locally compact Hausdorff space, then \(X\) is a Baire space: the intersection of countably many dense open subsets of \(X\) is dense in \(X\) [@Mun00].
A nonempty complete metric space is not a countable union of nowhere dense subsets.
Let \(X\) be a nonempty complete metric space and suppose \(X=\bigcup_{k\geq1}S_k\) with each \(S_k\) nowhere dense. Each \(U_k\coloneqq X\setminus\overline{S_k}\) is open, and it is dense because \(\overline{S_k}\) has empty interior. By the theorem, \(\bigcap_k U_k\) is dense in \(X\), hence nonempty since \(X\neq\emptyset\). But \(\bigcap_k U_k\subseteq X\setminus\bigcup_k S_k=\emptyset\), a contradiction.
State the Baire category theorem.