Continuous cofinal maps on ultrafilters

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.

💡 Research Summary

The paper introduces and studies the notion of “continuous Tukey reductions” for ultrafilters on a countable base. An ultrafilter U is said to have continuous Tukey reductions if, whenever an ultrafilter V is Tukey‑reducible to U, every monotone cofinal map f : U → V becomes continuous when restricted to some cofinal subset of U. The authors first formalize three increasingly strong continuity concepts: (1) continuous cofinal maps (existence of a continuous monotone cofinal map for every V ≤_T U), (2) continuous Tukey reductions (any monotone cofinal map becomes continuous on a cofinal sub‑ultrafilter), and (3) basic Tukey reductions (the restriction not only is continuous but can be extended to a continuous monotone map on the whole Cantor space 2^ω, generated by a level‑preserving, initial‑segment‑preserving map).

A central technical result (Lemma 7) shows that if a monotone cofinal map f : U → V is basic on a cofinal subset D⊆U, then f can be extended to a continuous monotone map \tilde f : 2^ω → 2^ω that is still basic and whose restriction to U remains cofinal into V. The proof constructs a level‑preserving map \hat g on all finite binary strings, using the basic generating map \hat f on D, and then defines \tilde f via unions of the images of finite approximations. This extension guarantees continuity because the construction respects the natural product topology on 2^ω.

The authors then identify a sufficient condition for the inheritance of continuous cofinal maps under Tukey reducibility (Theorem 8): if every monotone cofinal map on U has a cofinal domain on which it is basic, then any ultrafilter W ≤_T U automatically has basic Tukey reductions.

Applying this framework, the paper proves the “Main Theorem”: if an ultrafilter U is Tukey‑reducible to a p‑point (or, more generally, to a stable‑ordered‑union ultrafilter), then U possesses basic Tukey reductions. The proof relies on the known fact that p‑points have basic Tukey reductions (Theorem 4) and on the inheritance result of Theorem 8. Consequently, any ultrafilter below a p‑point inherits a strong form of continuity for all its Tukey‑reductions.

The second part of the paper extends these ideas to countable iterations of Fubini products of p‑points. A Fubini product of ultrafilters is a natural way to combine them, and iterating this operation yields highly structured ultrafilters. The authors model such iterated products using a topological Ramsey space of \vec{U}‑trees (denoted ~U‑trees). Each level of a ~U‑tree corresponds to a coordinate ultrafilter in the iteration, and a branch through the tree represents a choice of a set from each ultrafilter. By defining basic generating maps on each level and carefully synchronizing the level‑preserving sequences, they construct continuous monotone maps on the whole tree space that witness Tukey reductions. Theorem 9 (implicit in the text) asserts that any countable iteration of Fubini products of p‑points has basic Tukey reductions when viewed in the ~U‑tree topology.

These results have several notable consequences. First, they provide a new tool for analyzing ultrafilters added by σ‑closed forcings, because the existence of continuous cofinal maps often simplifies the study of the forcing extension’s ultrafilter structure. Second, they reveal situations where the Tukey and Rudin‑Keisler orders coincide: Theorem 5 (Raghavan) shows that if V is a q‑point and f : U → V is continuous, monotone, and cofinal, then V is Rudin‑Blass reducible to U, linking the two hierarchies. Finally, the paper demonstrates that the property of having basic Tukey reductions is, unlike many classical ultrafilter properties (e.g., being a p‑point or selective), robust under Tukey reducibility, thereby enriching the taxonomy of ultrafilter types.

In summary, the authors develop a robust framework for continuous Tukey reductions, prove that this property is inherited from p‑points and stable‑ordered‑union ultrafilters, and extend the theory to complex ultrafilters built from countable Fubini products via the machinery of ~U‑trees. The work deepens the connection between topological continuity, combinatorial ultrafilter properties, and the Tukey ordering, opening avenues for further research in both set‑theoretic topology and the structure theory of ultrafilters.