Supercompactness of compactifications and hyperspaces

Abstract:

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