Cofinal types of topological groups

[2605.25445] Cofinal types of topological groups Skip to main content Search Submit Donate Log in Mathematics > General Topology arXiv:2605.25445 (math) [Submitted on 25 May 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: Cofinal types of topological groups Authors: Xuan Gong , Dekui Peng View PDF HTML (experimental) Abstract: We investigate the local topological structure of non-metrizable topological groups through the lens of Tukey order and cofinal types. Motivated by recent advances in topological groups admitting an $\omega^\omega$-base, we introduce the \emph{fineness index}, denoted $\f(P)$, for arbitrary directed partially ordered sets. This cardinal invariant fundamentally generalizes the bounding number $\mathfrak{b}$ by capturing the exact threshold where a poset evades domination by its countable subsets, thereby establishing a universal lower bound for the character of topological groups with a $P$-base: $\chi(G) \in \{1, \omega\} \cup [fi(P), \text{cof}(P)]$. Furthermore, we resolve a structural problem regarding the exact cofinal types of free topological groups over uniform spaces. While classical results by Nickolas, Tkachenko, and others successfully computed the character of these groups via cardinal equalities (e.g., $\chi(F(X, \mathcal{U})) = \text{cof}(\mathcal{U}^\omega)$), lifting these equalities to strict Tukey equivalences has remained a persistent combinatorial challenge. By developing the novel machinery of \emph{neat trees} to refine uniform covering trees, we overcome the structural obstructions and prove the Tukey equivalence $\Ne_e(F(X, \U))=_T \U^\omega$ for any compact uniform space $(X, \U)$. Subjects: General Topology (math.GN) ; Group Theory (math.GR); Logic (math.LO) Cite as: arXiv:2605.25445 [math.GN] (or arXiv:2605.25445v2 [math.GN] for this version) https://doi.org/10.48550/arXiv.2605.25445 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Dekui Peng [ view email ] [v1] Mon, 25 May 2026 05:49:59 UTC (19 KB) [v2] Sat, 22 Aug 2026 03:52:21 UTC (20 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GN < prev | next > new | recent | 2026-05 Change to browse by: math math.GR math.LO References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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