The topology of ultrafilters as subspaces of $2^\omega$

TL;DR

This paper explores the topological properties of ultrafilters viewed as subspaces of the Cantor space, distinguishing them up to homeomorphism and constructing examples with specific homogeneity properties under Martin's Axiom.

Contribution

It introduces methods to classify non-principal ultrafilters by topological properties and constructs ultrafilters with countable dense homogeneity, addressing open questions.

Findings

Non-principal ultrafilters distinguished by topological properties

Existence of ultrafilters with countable dense homogeneous powers

Partial results relating topological and combinatorial properties

Abstract

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.