Consistently, Must Perfect Pseudocompact Spaces Be Separable?

Abstract

QUESTION: Is there a ZFC example of a pseudocompact space that is perfect but not separablc?In [3], it is shown that perfect pseudocompact spaces have the countable chain condition and are first countable. Furthermore, pseudocompact Moore spaces arc separable [4]. Because Moore spaces are perfect, we are prompted to consider whether perfect pseudocompact spaces have to be separable. It is not a ZFC theorem: a compact Suslin line is a counterexample. But is there a ZFC example?Associated consistency results can be found in [ 2 , 5 ] : assuming MA + TCH, every perfect (countably) compact space is separable-in fact, hereditarily separable. The space ? of [l] is a pseudocompact Moore space that is not hereditarily separable.If indccd there were an example answering this question, the results just given imply that it would be neither a Moore space nor locally compact.