Small MAD families whose Isbell-Mr\'owka spaces are pseudocompact

TL;DR

This paper constructs specific small maximal almost disjoint families in models of set theory whose associated Isbell-Mrówka spaces have pseudocompact Vietoris hyperspaces, demonstrating this property in various forcing extensions.

Contribution

It shows the existence of small MAD families with pseudocompact Isbell-Mrówka spaces in models of ZFC+CH and in certain forcing extensions, extending known results.

Findings

Existence of MAD families with pseudocompact Isbell-Mrówka spaces in models of ZFC+CH.

Construction of such families in Cohen extensions.

Identification of classical MAD families with pseudocompact hyperspaces.

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.