Pseudoradial spaces and copies of ω1 + 1

Abstract 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 = ω 2 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 → ω 1 + 1 in such a way that there is no copy of ω 1 + 1 in X that maps cofinally under f. We also give some conditions that imply the existence of copies of ω 1 in spaces. In particular, PFA implies that compact almost radial spaces of radial character ω 1 contain many copies of ω 1 .