Corollary

Every locally compact, metrizable space $X$ is Polish.

Let $X$ be the compact space obtained by adjoining a point at infinity to $X$; $X'$ is metrizable and of countable type ($§ 2,$ no. 9, Corollary to Proposition 16), and $X$ is complete with respect to its unique uniformity (Chapter II, $§ 4,$ no. 1, Theorem 1). Hence $X$ is a Polish space; since $X$ is open in $X'$, it follows that $X$ is Polish.