TITLE: If $X$ is not compact, does this mean that Cone($X$) is not compact?
QUESTION [5 upvotes]: I know that for $X$ a topological space, that if $X$ is compact then so is Cone($X$). Is the following identity also true:
If $X$ is non-compact then Cone($X$) is not compact, if not can somebody give a counterexample.
Thanks in advance.

REPLY [5 votes]: The canonical map $X\to C(X)$ identifies $X$ with a closed subset of $C(X)$. Therefore, if $C(X)$ is compact, then $X$ is compact.