Special sets of reals and weak forms of normality on Isbell-Mrówka spaces

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.

arXiv comment17 pages