Maximal almost disjoint families and pseudocompactness of hyperspaces

Abstract

We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if ${\mathsf{MA}}_{\mathfrak{c}}(\mathcal{P}(\omega )/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost disjoint family whose hyperspace is pseudocompact and prove that consistently such families do not exist \emph{genericaly}, by constructing a consistent example of a maximal almost disjoint family $\mathcal{A}$ of size less than $\mathfrak{c}$ whose hyperspace is not pseudocompact.