Corollary

Let X be a locally compact space and let K be a compact component of X. Then the neighbourhoods of K which are both open and closed form a fundamental system of neighbourhoods of K.

Let V be a relatively compact open neighbourhood of K in X (Chapter I, § 9, no. 7, Proposition 10) and let F be its frontier. Let U ⊂ \overline{V} be a set which is both open and closed with respect to \overline{V}. Then U is closed in X, and if in addition U does not meet F, then U is open in X (for then U ⊂ V and U is open in V). Hence it is enough to show that there is a subset of \overline{V} which contains K, does not meet F and is both open and closed with respect to \overline{V}.

Suppose this is not the case: then the intersections of F with the subsets of \overline{V} which contain K and are open and closed in \overline{V} form a filter base of closed sets in F; F is compact, so these sets have a common point y ∈ F. But this is absurd; since \overline{V} is compact, K is a component of \overline{V}, and by virtue of Proposition 6, the intersection of the sets which contain K and are both open and closed in \overline{V} is just K. The Corollary is thus proved.