  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, omega-covers, tau-covers, and gamma-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 unions,
arbitrary subsets, powers or products. Several interesting problems remain
open.
