If $X$ is compact and Hausdorff space, then vanishing at infinity implies compact support

Question

This is from Rudin.

The last sentence confused me. The inclusion relation is clear. But I can't see why they coincide if $X$ is compact.Any hint? Thanks.

Answer 1

If $X$ is compact, then every function on $X$ vanishes at infinity: just take $K = X$ and any function $f: X \rightarrow \mathbb{C}$ is vacuously less than any $\epsilon > 0$ for all of the precisely $0$ points in $X - K$. Hence, every complex function that is continuous on $X$ is in both $C_0(X)$ and $C_c(X)$. This is why Rudin identifies both with $C(X)$.