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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.