  The Stone-Cech compactification of the natural numbers bN, or equivalently,
the space of ultrafilters on the subsets of omega, is a well-studied space with
interesting properties. If one replaces the subsets of omega by partitions of
omega, one can define corresponding, non-homeomorphic spaces of partition
ultrafilters. It will be shown that these spaces still have some of the nice
properties of bN, even though none is homeomorphic to bN. Further, in a
particular space, the minimal height of a tree pi-base and P-points are
investigated.
