Theorem 3.13.4 (Urysohn’s Lemma (LCH)).label Let $X$ be a LCH space, $K\suf X$ be compact, $U\in\cali{N}(K)$, and $V$ as in Lemma 3.13.3 then there exists $F\in C_{c}(X;[0,1])$ such that $F|_{K}= 1,\supp{F}\suf \cl{V}$.
Proof. By Lemma 3.13.3, there exists $V,W\in\cali{N}^{o}(K)$ precompact such that
\begin{align*}K\suf V\suf\cl{V}\suf W\suf\cl{W}\suf U\end{align*}
As $\cl{W}$ is compact, it is normal by Proposition 3.10.6. Since $X$ is Hausdorff, $K\suf \cl{W}$ is closed.
By Urysohn’s Lemma, there exists $f\in C(\cl{W};[0,1])$ such that $f|_{K}=1$ and $f|_{\cl{W}\del V}=0$. Let
\begin{align*}F:X\to [0,1]&&x\mto \begin{cases}f(x)&x\in W\\ 0&x\in X\del \cl{V}\end{cases}\end{align*}
then $F|_{K}=1,\supp{F}\suf\supp{f}\suf \cl{V}$ and by Gluing Lemma for Continuous Functions $F\in C_{c}(X;[0,1])$.$\square$
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