The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality

Abstract

Abstract. Let I be an infinite pseudocompact space. We are interested in restrictions on k = \X\ and X -w(X) in addition to the obvious inequalities A < 2" and k < 2\ and k > 2", valid for X without isolated points (in particular for homogeneous X). We show that if cf(ic) -u, then X < 2<", and even X < 2'' for some n < k if X is homogeneous. Under the Singular Cardinals Hypothesis (which is much weaker than the GCH), there are no further restrictions for X without isolated points.

Countably compact group topologies on arbitrarily large free Abelian groups

“…E. van Douwen showed in [7] that the cardinality of a countably compact group cannot be a strong limit of countable cofinality.…”