Proposition 10

In a locally compact space $X$, every compact set $K$ has a fundamental system of compact neighbourhoods.

Let $U$ be any neighbourhood of $K$. For each $x \in K$ there is a compact neighbourhood $W(x)$ of $x$ contained in $U$. The interiors of the sets

W(x) form an open covering of K as x runs through K; hence there exist a finite number of points $x_i \in K$ ($1 \leq i \leq n$) such that the interiors of the W(x_i) cover K. The union V of the W(x_i) is therefore a compact neighbourhood of K contained in U (no. 3, Proposition 5).