The topology of ultrafilters as subspaces of 2ω

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martinʼs Axiom for countable posets, to distinguish non-principal ultrafilters on ω up to homeomorphism. Here, we identify ultrafilters with subpaces of 2ω in the obvious way. Using the same methods, still under Martinʼs Axiom for countable posets, we will construct a non-principal ultrafilter such that is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a P-point.