Countable π-character, countable compactness and PFA

Abstract

Balogh proved that PFA, the Proper Forcing Axiom, implies that a countably compact space with countable tightness is either compact or contains an uncountable free sequence. Eisworth established the relevance to proper forcing of strengthening the countable tightness assumption to that of hereditary countable π-character. Eisworth proved that for any countably compact space with hereditary countable π-character there is a totally proper forcing that adds an uncountable free sequence. We extend these results by showing that PFA implies that countably compact spaces are closed in spaces that have hereditary countable π-character. This gives a countably compact version of the Moore–Mrowka problem in the class of spaces with hereditary countable π-character.