Special sets of reals and weak forms of normality on Isbell-Mr\'owka spaces

TL;DR

This paper explores the relationships between various forms of normality in $ ext{Psi}$-spaces over almost disjoint families and special sets of reals, introducing a new class of sets that bridges known categories.

Contribution

It introduces a new class of special sets of reals corresponding to $ ext{aleph}_0$-separated families, situated between $ ext{lambda}$-sets and perfectly meager sets, and discusses potential almost-normality under forcing.

Findings

New class of special sets of reals introduced

Conditions for potential almost-normality in forcing extensions analyzed

Relationships between normality weakenings and special sets clarified

Abstract

We recall some classical results relating normality and some natural weakenings of normality in $\Psi$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $\lambda$-sets and $\sigma$-sets. We introduce a new class of special sets of reals which corresponds the corresponding almost disjoint family of branches being $\aleph_0$-separated. This new class fits between $\lambda$-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $\mathcal A$ being potentially almost-normal (pseudonormal), in the sense that $\mathcal A$ is almost-normal (pseudonormal) in some c.c.c. forcing extension.