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!