Weak $\lambda$-sets and almost-normal Isbell-Mrówka spaces

Abstract

We determine the least cardinality of a set of reals that is not a weak $\lambda$-set, proving that it is $\max \{\mathfrak{b},\mathfrak{s}\}$, where $\mathfrak{b}$ and $\mathfrak{s}$ are the bounding and splitting numbers. We prove that if $\mathfrak{s}<\mathfrak{b}$, then every first countable weak $\lambda$-space is a $\lambda$-space. We also show, relative to the consistency of $\mathsf{ZFC}$, that the existence of a weak $\lambda$-set that is not a $\lambda$-set is independent of $\mathsf{ZFC}+\mathfrak{b}=\mathfrak{s}$. In particular, we show that in the model obtained by adding ${\omega }_2$ Cohen reals over a model of $\mathsf{CH}$, every weak $\lambda$-set is a $\lambda$-set. Finally, we construct an almost-normal MAD family under $\mathsf{CH}$ and show that no such family exists under ${\mathsf{OCA}}_{\mathrm{T}}+\mathfrak{ap}>{\omega }_1$. Thus, the existence of an almost-normal MAD family is independent of $\mathsf{ZFC}$.

arXiv comment27 pages, 2 figures