Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $\beta \omega$

Abstract

We study the relations between a generalization of pseudocompactness, named $(\kappa, M)$-pseudocompactness, the countably compactness of subspaces of $\beta \omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\mathfrak c$-many selective ultrafilters, that there exists a subspace of $\beta \omega$ that is $(\kappa, \omega^*)$-pseudocompact for all $\kappa<\mathfrak c$, but $\text{CL}(X)$ isn't pseudocompact. We prove in ZFC that if $\omega\subseteq X\subseteq \beta\omega$ is such that $X$ is $(\mathfrak c, \omega^*)$-pseudocompact, then $\text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $\mathfrak c$ for some small cardinals. We provide an example of a subspace of $\beta \omega$ for which all powers below $\mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $\omega \subseteq X$, the pseudocompactness of $\text{CL}(X)$ implies that $X$ is $(\kappa, \omega^*)$-pseudocompact for all $\kappa<\mathfrak h$, and provide an example of such an $X$ that is not $(\mathfrak b, \omega^*)$-pseudocompact.