Corollary

Let $X$ be a locally compact space and let $X'$ be the compact space obtained by adjoining a point at infinity $\omega$ to $X$ (Chapter I, § 9, no. 8). Then the following statements are equivalent:

a) The topology of $X$ has a countable base. b) $X'$ is metrizable. c) $X$ is metrizable and $\sigma$-compact.

$a \Longrightarrow b$: Let $U_n$ be a countable base of the topology of $X$. Each neighbourhood of a point $x \in X$ contains a compact neighbourhood of $x$, which in turn contains a neighbourhood of $x$ equal to some $U_n$. Hence the relatively compact $U_n$ form a base of the topology of $X$, and we may therefore suppose that all the $U_n$ are relatively compact. $X$ is therefore a countable union of compact sets $\overline{U}_n$, i.e. it is $\sigma$-compact; this implies that in $X'$ the point $\omega$ has a countable fundamental system $V_n$ of open neighbourhoods (Chapter I, § 9, no. 9, Proposition 15, Corollary 2). Hence each neighbourhood of a point $y \in X'$ contains either one of the $U_n$ or one of the $V_n$, which is a neighbourhood of $y$, and so the $U_n$ and the $V_n$ form a countable base of the topology of $X'$. Hence $X'$ is metrizable, by Proposition 16.

b) $\Longrightarrow c)$: If $X'$ is metrizable, then $\omega$ has a countable fundamental system of neighbourhoods, and therefore $X$ is $\sigma$-compact by Chapter I, § 9, no. 9, Proposition 15, Corollary 2.

c) $\Longrightarrow a)$: By hypothesis, there is an increasing sequence $(V_n)$ of relatively compact open sets which cover $X$ and are such that $\overline{V}_n \subset V_{n+1}$ (Chapter I, § 9, no. 9, Proposition 15). The subspace $\overline{V}_n$ is compact and metrizable and therefore has a countable base (Proposition 16), and therefore so does $V_n$. Let $(U_{mn})_{m \geq 1}$ be a base of the topology of $V_n$. For each $x \in X$ and each neighbourhood $W$ of $x$, there exists $n$ such that $x \in V_n$, hence there exists $m$ such that $x \in U_{mn} \subset V_n \cap W$. Hence the sets $U_{mn}$ ($m \geq 1, n \geq 1$) form a base of the topology of $X$.