TITLE: $C_c(X)$ is complete, then $X$ is compact
QUESTION [10 upvotes]: Let $X$ be a locally compact Hausdorff space such that $C_c(X),$ the space of all continuous functions with compact support is complete. Show that $X$ is compact. 

My attempt:
I have shown that $C_c(X)$ is dense in $C_0(X),$ the space of all continuous functions vanishing at infinity. Since $C_c(X)$ is complete, therefore $$C_c(X)=C_0(X).$$
Now to conclude that $X$ is compact, it suffices to find a function in $C_0(X)$ which vanishes nowhere on $X.$ Is this always possible?

REPLY [3 votes]: If $\Omega$ is $\sigma$-compact, then the space is always complete when endowed with the strict inductive limit topology of Fréchet spaces. You have to specify which topology you are considering on that space.