Pseudoradial spaces and copies of ω1 + 1

In this paper we compare the concepts of pseudoradial spaces and the recently defined strongly pseudoradial spaces in the realm of compact spaces. We show that MA+c=ω2 implies that there is a compact pseudoradial space that is not strongly pseudoradial. We essentially construct a compact, sequentially compact space X and a continuous function f:X→ω1+1 in such a way that there is no copy of ω1+1 in X that maps cofinally under f. We also give some conditions that imply the existence of copies of ω1 in spaces. In particular, PFA implies that compact almost radial spaces of radial character ω1 contain many copies of ω1.