Theorem 3.13.5 (Tietze Extension Theorem (LCH)).label Let $X$ be a LCH space, $K\suf X$ compact, $U\in\cali{N}^{o}(K)$, $f\in C(K;\R)$, and $V$ as in Lemma 3.13.3 then there exists $F\in C_{c}(X;\R)$ such that $F|_{K}=f,\supp{F}\suf \cl{V}$ .
Proof. By Lemma 3.13.3, there exists $V,W\in\cali{N}^{o}(K)$ precompact such that
As $\cl{W}$ is compact, it is normal by Proposition 3.10.6. Since $X$ is Hausdorff, $K\suf \cl{W}$ is closed.
By Tietze Extension Theorem, there exists $g\in C(\cl{W};\R)$ such that $g|_{K}=f$. By Urysohn’s Lemma, there exists $\eta\in C_{c}(X;[0,1])$ such that $\eta|_{K}=1,\supp{\eta}\suf \cl{V}$. Let
then $F|_{K}=g|_{K}=f,\supp{F}\suf\supp{\eta}\suf \cl{V}$ and by Gluing Lemma for Continuous Functions $F\in C_{c}(X;\R)$.$\square$
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