The topology of ultrafilters as subspaces of $2^ω$

Details The topology of ultrafilters as subspaces of $2^ω$ The topology of ultrafilters as subspaces of $2^ω$

2011-08-11

arxiv.org/abs/1108.2533v2

Mathematics

General Topology

22 pages

Scientific paper

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $\omega$ up to homeomorphism. Here, we identify ultrafilters with subpaces of $2^\omega$ in the obvious way. Using the same methods, still under Martin's Axiom for countable posets, we will construct a non-principal ultrafilter $\UU\subseteq 2^\omega$ such that $\UU^\omega$ is countable dense homogeneous. This consistently answers a question of Hru\v{s}\'ak and Zamora Avil\'es. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a $\mathrm{P}$-point.

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