Supercompactness of compactifications and hyperspaces

Abstract

We prove a theorem which implies that if γ ω \gamma \omega is a supercompact compactification of the countable discrete space ω \omega then γ ω − ω \gamma \omega - \omega is separable. This improves an earlier result of the author’s that such a γ ω \gamma \omega must have γ ω − ω ccc \gamma \omega - \omega \;{\text {ccc}} . We prove a theorem which implies that the hyperspace of closed subsets of 2 ω 2 {2^{\omega _2}} is not a continuous image of a supercompact space. This improves an earlier result of L {\text {L}} . Šapiro that the hyperspace of closed subsets of 2 ω 2 {2^{\omega _2}} is not dyadic.