# Extensions of functions vanishing at infinity on locally compact Hausdorff spaces

working myself through a proof of the fact, that every closed Ideal in $C_0(X)$ for a locally compact Hausdorff space X is of the form $I_F=\{f\in C_0(X)\colon f\lvert_F \equiv 0\}$ for one and only one closed subspace $F\subseteq X$, I came to the point where I want to show that for every $F$ as above the following holds $$I_F\cong C_0(X\setminus F).$$

Since the canonical inclusion $I\colon F \to X$ is proper, $i_F^\ast\colon C_0(X) \to C_0(F)$ is a *-homomorphism.

Then I want to use some kind of argument involving Tietze's theorem to show that this map is surjective I guess. But since Tietze's extension theorem applies only to normal spaces and to my knowledge locally compact spaces needn't be normal I'm a bit confused here. Can someone point me to a corollary that does the trick?

Thanks!

I don't think you need any theorem. If $f\in I_F$, then you can see $f\in C_0(X\setminus F)$ by restriction. And if $g\in C_0(X\setminus F)$, extend to $C_0(X)$ by $g|_F=0$.