Can I apply Urysohn's lemma on a locally compact Hausdorff space?

Question

Let $\Omega$ be a locally compact Hausdorff space. Suppose that $A$ and $B$ are disjoint closed subsets. Does there exist a continuous map $f\colon\Omega\to\mathbb{R}$ such that $f[\Omega]\subset[0,1]$, $f|_{A}\equiv0$ and $f|_{B}\equiv1$? I know that it is possible for normal topological spaces.

Answer 1

No. Urysohn's lemma holds if and only if the space is normal. However not every locally compact Hausdorff space is normal.

(Check here: https://mathoverflow.net/questions/53300/locally-compact-hausdorff-space-that-is-not-normal)

Nevertheless, there is a version of Urysohn's lemma that holds in your case: if $\Omega$ is a locally compact Hausdorff space and $A$, $B$ are disjoint subsets of $\Omega$ such that $A$ is compact and $B$ is closed, then there exists a continuous map $f:\Omega\to [0,1]$ such that $f|_A\equiv 0$ and $f|_B\equiv 1$.

(Check here: https://www.math.ksu.edu/~nagy/real-an/1-05-top-loc-comp.pdf)