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

Abstract:

Let X be an infinite pseudocompact space. We are interested in restrictions on $\kappa = |X|$ and $\lambda = w(X)$ in addition to the obvious inequalities $\lambda \leqslant {2^\kappa }$ and $\kappa \leqslant {2^\lambda }$ and $\kappa \geqslant {2^\omega }$, valid for X without isolated points (in particular for homogeneous X). We show that if ${\text {cf}}(\kappa ) = \omega$ then $\lambda \leqslant {2^{ < \kappa }}$, and even $\lambda \leqslant {2^\mu }$ for some $\mu < \kappa$ 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.