Lemma 3.13.3.label Let $X$ be a LCH space, $K\suf X$ be compact, and $U\in\cali{N}(K)$, then there exists $V\in\cali{N}^{o}(K)$ precompact such that $K\suf V\suf\cl{V}\suf U$. In particular $X$ is regular.
Proof. For each $x\in K$, there exists $V_{x}\in\cali{N}^{o}(x)$ such that $x\in V_{x}\suf\cl{V_x}\suf U$ by Definition 3.13.1(3). Since $K$ is compact there exists $\curl{x_j}_{1}^{n}\suf K$ such that $K\suf \cups{n}{j=1}V_{x_j}$ hence
\begin{align*}K\suf \cl{\cups{n}{j=1}V_{x_j}}=\cups{n}{j=1}\cl{V_{x_j}}\suf U\end{align*}
we can take $V=\cups{n}{j=1}V_{x_j}\in\cali{N}^{o}(K)$.$\square$
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