TITLE: Does $C_0(X)$ determine the topology for a locally compact space $X$?
QUESTION [6 upvotes]: Given a locally compact Hausdorff space, does $C_0(X)$, the continuous functions vanishing at infinity, determine the topology of $X$?
For example, for a net $\{x_{\alpha}\}\subset X$ if I have $f(x_\alpha)\to f(x)$ for all $f\in C_0(X)$, does it follow that $x_{\alpha} \to x$ ?
I cannot find any reference to the Banach-Stone theorem that proves this. I would really appreciate some feedback.
Thanks!

REPLY [2 votes]: Let $X^\ast$ be the one-point compactification of $X$, with compactifying point $\infty$, then if $C$ is closed in $X$ and $x \in X\setminus C$, note that $C^\ast:=C \cup \{\infty\}$ is compact in $X^\ast$ and as $X^\ast$ is normal we can find a continuous $f: X^\ast \to \mathbb{R}$ such that $f(x)=1$ and $f[C^\ast]=\{0\}$. And then $g=f\restriction X \in C_0(X)$ and $g(x) \notin \overline{g[C]}$ and so the set $C_0(X)$ separates points and closed sets. By well-known general topology facts means that it determines the topology on $X$ (the topology on $X$ is the unique smallest topology that makes all functions in $C_0(X)$ continuous) and what you state about nets is a consequence of that fact.