The combinatorics of splittability

Abstract

Marion Scheepers, in his studies of the combinatorics of open covers, introduced the property Split$\text{(}\text{U}\text{,}\text{V}\text{)}$ asserting that a cover of type $\text{U}$ can be split into two covers of type $\text{V}$. In the first part of this paper we give an almost complete classification of all properties of this form where $\text{U}$ and $\text{V}$ are significant families of covers which appear in the literature (namely, large covers, ω-covers, τ-covers, and γ-covers), using combinatorial characterizations of these properties in terms related to ultrafilters on $\text{N}$.

In the second part of the paper we consider the questions whether, given $\text{U}$ and $\text{V}$, the property Split$\text{(}\text{U}\text{,}\text{V}\text{)}$ is preserved under taking finite or countable unions, arbitrary subsets, powers or products. Several interesting problems remain open.