The combinatorics of splittability

Marion Scheepers, in his studies of the combinatorics of open covers, introduced the property Split(U,V) asserting that a cover of type U can be split into two covers of type V. In the first part of this paper we give an almost complete classification of all properties of this form where U and 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 ℕ. In the second part of the paper we consider the questions whether, given U and V, the property Split(U,V) is preserved under taking finite or countable unions, arbitrary subsets, powers or products. Several interesting problems remain open.