Continuous cofinal maps on ultrafilters

TL;DR

This paper investigates conditions under which ultrafilters have continuous Tukey reductions, showing inheritance under Tukey reducibility and continuity in Fubini product iterations of p-points.

Contribution

It establishes that ultrafilters Tukey reducible to p-points have continuous Tukey reductions and demonstrates continuity properties in Fubini product iterations of p-points.

Findings

Ultrafilters Tukey reducible to p-points have continuous Tukey reductions.

Countable Fubini product iterations of p-points have continuous Tukey reductions.

Continuity of Tukey reductions is inherited under certain conditions.

Abstract

An ultrafilter $\mathcal{U}$ on a countable base {\em has continuous Tukey reductions} if whenever an ultrafilter $\mathcal{V}$ is Tukey reducible to $\mathcal{U}$, then every monotone cofinal map $f:\mathcal{U}\ra\mathcal{V}$ is continuous when restricted to some cofinal subset of $\mathcal{U}$. In the first part of the paper, we give mild conditions under which the property of having continuous Tukey reductions is inherited under Tukey reducibility. In particular, if $\mathcal{U}$ is Tukey reducible to a p-point then $\mathcal{U}$ has continuous Tukey reductions. In the second part, we show that any countable iteration of Fubini products of p-points has Tukey reductions which are continuous with respect to its topological Ramsey space of $\vec{\mathcal{U}}$-trees.