Lemma 4.20.2. Let $X$ be a LCH space, $K \subset X$ be compact, and $U \in \cn(K)$, then there exits $V \in \cn^{o}(K)$ precompact such that $K \subset V \subset \ol{V}\subset U$.
Proof. For each $x \in K$, there exists $V_{x} \in \cn^{o}(x)$ be precompact such that $x \in V_{x} \subset \overline{V_x}\subset U$ by (3) of Definition 4.20.1. Since $K$ is compact, there exists $\seqf{x_j}\subset K$ such that
\[K \subset \bigcup_{j = 1}^{n} V_{x_j}\subset U\]
\[\ol{\bigcup_{j = 1}^n V_{x_j}}= \bigcup_{j = 1}^{n} \overline{V_{x_j}}\subset U\]
so $V = \bigcup_{j = 1}^{n} V_{x_j}\in \cn^{o}(K)$ is precompact.$\square$