The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. That is, if $f$ is continuous then there exists a point $c$ in $X$ such that $f(c)= \sup \{f(x) : x \in X \}$, where this supremum is necessarily finite.
My question is: Does the converse of this theorem hold in general? That is, if every continuous function on a space $X$ has the extreme value property, then is $X$ compact? Or is it instead true that there exists a topological space $X$ which is not compact but such that every continuous function on $X$ has the extreme value property? If the latter is true, I would like an example. If the former is true, a proof would be ideal.
Thanks!
