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.
