TITLE: Do non-compact spaces always contain "big" closed subsets?
QUESTION [5 upvotes]: I was wondering if the following is true or not:

Let $X$ be a non-compact topological space. Does there exist a closed non-compact subset $C$ such that $C\neq X$?

The result is obvious if, for example, $X$ has an isolated point $x$ because then $X\setminus\{x\}$ is closed. But what if that is not the case?
Thank you.

REPLY [5 votes]: Let $\{U_\alpha\}$ be an open cover of $X$ by nonempty open sets with no finite subcover, and pick an arbitrary $U$ from this cover. Then $C=X-U$ is a noncompact proper closed subset of $X$. The same cover of $C$ has no finite subcover for otherwise we could add $U$ to form a finite subcover of $X$.