Maximal countably compact spaces and embeddings in MP-spaces

摘要

We study embeddings in maximal pseudocompact spaces together with maximal countable compactness in the class of Tychonoff spaces. It is proved that under MA CH any compact space of weight is a retract of a compact maximal pseudocompact space. If kappa is strictly smaller than the first weakly inaccessible cardinal, then the Tychonoff cube [0, 1](kappa) is maximal countably compact. However, for a measurable cardinal kappa, the Tychonoff cube of weight kappa is not even embeddable in a maximal countably compact space. We also show that if X is a maximal countably compact space, then the functional tightness of X is countable. It is independent of ZFC whether every compact space of countable tightness must be maximal countably compact. On the other hand, any countably compact space X with the Mazur property ( every real-valued sequentially continuous function on X is continuous) must be maximal countably compact. We prove that for any omega-monolithic compact space X, if C (p) (X) has the Mazur property, then it is a Fr,chet-Urysohn space.