(Question about the proof of $M(C_0(X))=C_b(X)$ )- Urysohn's lemma for locally compact Hausdorff spaces

Question

I'm stuck on understanding the proof of why the multiplier algebra of $C_0(X)$ can be identified with $C_b(X)$, $X$ is a locally compact Hausdorff space.

The proof uses that $C_0(X)$ is an essentiell ideal in $C_b(X)$ so as to apply the universal property of multiplier algebras. With Urysohn's lemma you obtain that this ideal is essentiell. But which version of Urysohn's lemma you need here? Because there are locally compact Hausdorff space X, which are not normal.

Regards.

Answer 1

Urysohn's Lemma requires precisely that $X$ be locally compact and Hausdorff. See, for instance, 2.12 in Rudin's Real and Complex Analysis.

This works because compact and Hausdorff implies normal; so, locally, a locally compact Hausdorff space behaves like a normal one.