Corollary 2

*Let $X$ be a locally compact space and let $X'$ be the compact space obtained by adjoining a point at infinity $\omega$ to $X$ (no. 8). Then $X$ is $\sigma$-compact if and only if the point $\omega$ has a countable fundamental system of neighbourhoods in $X'$.

If $X$ is $\sigma$-compact we can construct a sequence of subsets $U_n$ of $X$ as in Proposition 15, and the neighbourhoods $X' - \overline{U}_n$ of $\omega$ in $X'$ form a fundamental system of neighbourhoods of $\omega$ by reason of Corollary 1. The converse follows from the fact that the complements of open neighbourhoods of $\omega$ are compact subsets of $X$.

Clearly every closed subspace of a locally compact $\sigma$-compact space is locally compact and $\sigma$-compact. Likewise any finite product of locally compact $\sigma$-compact spaces is locally compact and $\sigma$-compact.

Notice, however, that an open subspace of a compact space need not be $\sigma$-compact, as Alexandroff's theorem (no. 8, Theorem 4) shows.