Topological properties of incomparable families
Volume 156 / 2019
Abstract
We study topological properties of families of mutually incomparable subsets of $\omega $. We say that two subsets $a$ and $b$ of $\omega $ are incomparable if both $a\setminus b$ and $b\setminus a$ are infinite. We raise the question whether there may be an analytic maximal incomparable family and show that (1) it cannot be $K_\sigma $, and (2) every incomparable family with the Baire property is meager. On the other hand, we show that non-meager incomparable families exist in $\mathsf{ZFC}$, while the existence of a non-null incomparable family is consistent. Finally, we show that there are maximal incomparable families which are both meager and null assuming either $\mathfrak r=\mathfrak c$ or the existence of a completely separable MAD family; in particular they exist if $\mathfrak c \lt \aleph _\omega $. Assuming {\sf CH}, we can even construct a maximal incomparable family which is concentrated on a countable set, and hence of strong measure zero.
