# Urysohn lemma for locally compact spaces

How can I prove the following theorem?

Let $X$ locally compact, $K \subseteq X$ compact, $O \subseteq X$ open with $K \subseteq O$. Then there exists $f \in C_c(X)$ with $0 \leq f \leq 1$, $f|_K \equiv 0$ and $\operatorname{supp}(f) \subseteq O$.

(Where $C_c(X)=\{f ∈ C(X, \mathbb C) : \operatorname{supp}(f) \text{ compact}\}$).

I think that I would need the topological lemma for locally compact spaces, but I couldn't come up with a suitable proof. Thanks in advance!

• The Tychonoff plank is a standard example for what you stated. It's $((\omega_1 + 1) \times (\omega+1)) \setminus \{(\omega_1,\omega)\}$, where the ordinal numbers have the order topology etc. – Henno Brandsma Jun 10 '16 at 12:08