Madness and (weak) normality

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 García-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

19 pages