Theorem 1 (Baire)

(i) Every locally compact space $X$ is a Baire space. (ii) Every topological space $X$ on which there exists a metric, compatible with the topology of $X$ and which defines on $X$ the structure of a complete metric space, is a Baire space.

We shall show that axiom (EB) is satisfied in each case. Let $(A_n)$ be a sequence of dense open sets in $X$, and let $G$ be any non-empty open set. We can then define inductively a sequence $(G_n)$ of non-empty open sets such that $G_1 = G$ and $\overline{G}_{n+1} \subset G_n \cap A_n$: for since by hypothesis $G_n$ is not empty, $G_n \cap A_n$ is a non-empty open set; and since $X$ is regular in both cases envisaged, there is a non-empty open set $G_{n+1}$ such that $\overline{G}_{n+1} \subset G_n \cap A_n$. Hence the set $G \cap \bigcap_{n=1}^\infty A_n$ contains the intersection of the sets $G_n$, which is equal to the intersection of the sets $\overline{G}_n$; hence it is enough to show that $\bigcap_{n=1}^\infty \overline{G}_n \neq \varnothing$. Now if $X$ is locally compact, we may assume that $\overline{G}_2$ is compact; in the compact space $\overline{G}_2$, the $\overline{G}_n (n \geq 2)$ form a decreasing sequence of non-empty closed sets, and their intersection is therefore not empty by axiom (C'') (cf. Chapter I, § 9, no. 1, Definition 1]. If $X$ is a complete metric space (with respect to a metric compatible with its topology) we may suppose that $\overline{G}_n$ has been chosen so that its diameter (with respect to this metric) tends to 0 as $n$ tends to $+\infty$; the $\overline{G}_n$ therefore form a Cauchy filter base which converges to a point $x$, and $x$ necessarily belongs to the intersection of the sets $\overline{G}_n$.

Q.E.D.