# Countably compact non-compact perfect spaces

Recall that a space is countably compact if every infinite set has an accumulation point. A space is perfect if every closed set is a countable intersection of open sets. One of the classical applications of Martin's Axiom to topology is the following old theorem due to Bill Weiss.

Under $$MA+\neg CH$$ every countably compact perfect regular space is compact.

The above theorem cannot be proved in ZFC since, under Jensen's Axiom $$\diamondsuit$$, for example, there is a countably compact perfectly normal non-compact space (Ostaszewski space).

Regularity of the space is essential in Weiss's argument, but is it actually needed in the above theorem?

QUESTION: Is it true under $$MA+\neg CH$$ that every countably compact perfect Hausdorff space is compact?

At least if one strengthens $$MA+\neg CH$$ to the Proper Forcing Axiom then regularity is not needed anymore. Indeed, if $$X$$ is a perfect countably compact space then $$X$$ does not contain any uncountable discrete sets, and that along with PFA implies that $$X$$ is Lindelof. Now every Lindelof countably compact space is compact.