Madness and (weak) normality

TL;DR

This paper explores weakenings of normality in $ ext{Psi}$-spaces, establishing independence results, constructing specific examples, and clarifying relationships between different normality concepts in set-theoretic topology.

Contribution

It proves the independence of the existence of certain almost-normal $ ext{Psi}$-spaces from ZFC, constructs a partly-normal not quasi-normal AD family, and distinguishes between almost-normal and strongly $ ext{aleph}_0$-separated AD families.

Findings

Existence of a MAD family with almost-normal $ ext{Psi}$-space is independent of ZFC.

Constructed a partly-normal not quasi-normal AD family.

Showed almost-normal and strongly $ ext{aleph}_0$-separated AD families are different, even under CH.

Abstract

We consider weakenings of normality in $\Psi$-spaces and prove that the existence of a MAD family whose $\Psi$-space is almost-normal is independent of \textsf{ZFC}. We also construct a partly-normal not quasi-normal AD family, answering questions from Garc\'ia-Balan and Szeptycki. We finish by showing that the concepts of almost-normal and strongly $\aleph_0$-separated AD families are different, even under \textsf{CH}, answering a question from Oliveira-Rodrigues and Santos-Ronchim