Small MAD families whose Isbell--Mr\'owka space has pseudocompact hyperspace

Abstract

Given a countable transitive model $M$ for ZFC+CH, we prove that one can produce a maximal almost disjoint family in $M$ whose Vietoris Hyperspace of its Isbell-Mr\'owka space is pseudocompact on every Cohen extension of $M$. We also show that a classical example of $\omega_1$-sized maximal almost disjoint family obtained by a forcing iteration of length $\omega_1$ in a model of non CH is such that the Vietoris Hyperspace of its Isbell-Mr\'owka space is pseudocompact.