Abstract
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 U⊆2ω such that Uω 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.
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