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Let $X$ be a locally compact Hausdorff space, $K \subseteq X$ compact, $V_1, V_2 \subseteq X$ open s.t. $K \subseteq V_1 \cup V_2$.

According to this proof of Riesz' representation theorem (step 1) it follows from Urysohn's lemma, that there are compactly supported continuous $h_i : X \to [0,1]$ s.t. $\text{supp}(h_i) \subseteq V_i$ and $h_1(x)+h_2(x) = 1$ for all $x \in K$.

Does anyone see the argument? I know the variant of Urysohn's lemma for locally compact Hausdorff spaces but I don't see at all how it should help me here.

Thanks in advance!

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One form of Urysohn's lemma for LCH spaces says that, given compact $L$ contained in open $U$, you can find a continuous function $f$ into $[0,1]$ which is compactly supported inside $U$ such that $f|_L=1$. Then, apply this with $L:=K$ and $U:=V_1\cup V_2$ to give $f$, supported on compact $M$. You then have compact $M\setminus V_1$ contained in open $V_2$, so by local compactness you can find compact $N$ contained between $M\setminus V_1$ and $V_2$ such that $N$ contains a neighborhood of $M\setminus V_1$. Then apply Urysohn's lemma again with $L:=N$ and $U:=V_2$ to get $g$. Set $h_1:=f(1-g), h_2:=fg$.

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