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 to 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.
Keywords: Isbell-Mrówka spaces; almost disjoint families; almost-normal; weak \lambda-set
DOI: DOI 10.14712/1213-7243.2023.014
AMS Subject Classification: 54D15 54D80