Spaces for which compactness is equivalent to pseudocompactness

Question

A metric space is compact if and only if it is pseudocompact.

Can we minimize metric space condition?

Answer 1

For a normal ($T_4$) space we have that pseudocompactness is equivalent to countable compactness (in the sense that a countable open cover has a finite subcover, which is, for that class of spaces, also equivalent to limit point compactness (any infinite set has a limit point (or even an $\omega$-limit point)). We cannot get full compactness easily (demanding Lindelöf is a boring way to do that), as shown by examples as the long line, which is locally metrisable, orderable (so hereditarily normal), countably compact and pseudocompact etc. but not compact.

One nice way we could try to properly weaken the class of metric spaces is to go to perfectly normal ones (normal and closed sets are $G_\delta$ sets). But then we get consistency results only: in a model of ZFC where $\textrm{MA}(\omega_1)$ holds, indeed perfectly normal pseudocompact spaces are compact, but if $\diamondsuit$ holds (as in V=L) we have so-called Ostaszewski spaces which are perfectly normal, hereditarily separable, countably compact, locally compact etc. but non-compact.