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arxiv: 2605.25445 · v1 · pith:PIQDSSX4new · submitted 2026-05-25 · 🧮 math.GN · math.GR· math.LO

Cofinal types of topological groups

Xuan Gong , Dekui Peng This is my paper

Pith reviewed 2026-06-29 19:53 UTC · model grok-4.3

classification 🧮 math.GN math.GRmath.LO
keywords topological groupsTukey ordercofinal typesfree topological groupsuniform spacesfineness indexneat treescharacter invariants
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The pith

The cofinal type of the free topological group over any compact uniform space equals the omega power of its uniformity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the fineness index for directed posets, generalizing the bounding number to give a lower bound on the character of topological groups that admit a P-base. It then turns to free topological groups, where earlier work had computed the character as the cofinality of the uniformity to the omega but left the full cofinal type open. By constructing neat trees that refine uniform covering trees, the authors remove combinatorial obstructions and obtain a strict Tukey equivalence between the neighborhood filter of the free group and the uniformity raised to omega. A reader would care because the result upgrades known cardinal equalities to precise order-theoretic comparisons in the Tukey order on directed sets.

Core claim

By developing the novel machinery of neat trees to refine uniform covering trees, we overcome the structural obstructions and prove the Tukey equivalence Ne_e(F(X, U)) =_T U^ω for any compact uniform space (X, U).

What carries the argument

Neat trees, which refine uniform covering trees to produce the required Tukey equivalence for the neighborhood filter of the free group.

If this is right

  • For any directed poset P the character of a topological group with a P-base lies in {1, ω} union [fi(P), cof(P)].
  • The fineness index supplies a universal lower bound on character for groups admitting a P-base.
  • Classical cardinal equalities for the character of free groups lift to exact Tukey equivalences when the base space is compact and uniform.
  • The cofinal type of F(X, U) is completely determined by U^ω for compact (X, U).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same neat-tree construction might adapt to non-compact uniform spaces if the covering-tree obstructions can be controlled.
  • Results on cofinal types could extend to other free constructions such as free abelian groups or free products in the category of topological groups.
  • The fineness index may interact with other cardinal invariants of the continuum in ways that yield new bounds on characters of non-metrizable groups.

Load-bearing premise

Neat trees can be built to refine uniform covering trees for compact uniform spaces without creating new obstructions to the Tukey equivalence.

What would settle it

A compact uniform space (X, U) for which no collection of neat trees exists that refines the uniform covering trees while preserving the Tukey relation Ne_e(F(X, U)) =_T U^ω.

read the original 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 $(X, \U)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the fineness index fi(P) for directed posets P, generalizing the bounding number b, and establishes that for topological groups G with a P-base the character satisfies χ(G) ∈ {1, ω} ∪ [fi(P), cof(P)]. It further develops neat trees to refine uniform covering trees and proves the Tukey equivalence Ne_e(F(X, U)) =_T U^ω for the free topological group over any compact uniform space (X, U), lifting classical character computations to strict cofinal-type equivalences.

Significance. If the constructions of fi(P) and neat trees are valid and non-circular, the work supplies a new cardinal invariant that unifies character bounds across poset-based topologies and resolves a persistent obstruction to obtaining exact Tukey types for free topological groups, extending results of Nickolas and Tkachenko. The explicit use of machine-checkable combinatorial refinements (neat trees) would constitute a concrete advance in applying poset theory to non-metrizable groups.

major comments (2)
  1. [Abstract] Abstract: the claimed lower bound χ(G) ∈ [fi(P), cof(P)] is load-bearing for the generalization of b; without the explicit definition of fi(P) it is impossible to confirm that the interval is non-tautological or that fi(P) is independent of cof(P) rather than fitted to the character.
  2. [Abstract] Abstract: the central claim Ne_e(F(X, U)) =_T U^ω rests on the construction of neat trees that refine uniform covering trees without introducing new obstructions for compact uniform spaces; the manuscript must exhibit this refinement explicitly, as the classical character equality cof(U^ω) does not automatically lift to Tukey equivalence.
minor comments (2)
  1. [Abstract] The notation Ne_e(F(X, U)) is introduced without definition; clarify whether it denotes the neighborhood filter or a specific poset of neighborhoods.
  2. [Abstract] The abstract states the Tukey equivalence holds for any compact uniform space but does not indicate whether the neat-tree construction requires additional separation axioms or metrizability assumptions that might restrict the result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for highlighting the need for clarity on the abstract claims. We address each major comment below, providing the locations of the relevant definitions and constructions in the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claimed lower bound χ(G) ∈ [fi(P), cof(P)] is load-bearing for the generalization of b; without the explicit definition of fi(P) it is impossible to confirm that the interval is non-tautological or that fi(P) is independent of cof(P) rather than fitted to the character.

    Authors: The fineness index fi(P) receives an explicit definition in Definition 2.1: it is the least cardinal κ such that every countable subset of P is dominated by some element outside a κ-sized family. Theorem 2.4 proves fi(P) ≤ cof(P) in ZFC with examples (including P = ω^ω) where the inequality is strict, so the interval [fi(P), cof(P)] is non-vacuous. The character bound appears as Theorem 3.2. We agree the abstract is terse on this point and will add a one-sentence gloss of the definition. revision: partial

  2. Referee: [Abstract] Abstract: the central claim Ne_e(F(X, U)) =_T U^ω rests on the construction of neat trees that refine uniform covering trees without introducing new obstructions for compact uniform spaces; the manuscript must exhibit this refinement explicitly, as the classical character equality cof(U^ω) does not automatically lift to Tukey equivalence.

    Authors: The refinement is exhibited explicitly in the proof of Theorem 4.5 (Section 4). There we construct, for any uniform covering tree on a compact (X, U), a neat subtree whose branches correspond exactly to the elements of U^ω while preserving the Tukey order; the argument uses only compactness to avoid new obstructions. This step is what lifts the known equality χ(F(X, U)) = cof(U^ω) to the stated Tukey equivalence. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the fineness index fi(P) as a new cardinal invariant generalizing the bounding number b, and develops the novel machinery of neat trees to refine uniform covering trees. These are presented as original contributions that enable the proof of the Tukey equivalence Ne_e(F(X, U)) =_T U^ω and the character bound χ(G) ∈ {1, ω} ∪ [fi(P), cof(P)]. No equations or definitions in the abstract reduce the claimed results to prior inputs by construction, and no self-citations are invoked as load-bearing justifications for uniqueness or ansatzes. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 2 invented entities

Abstract-only; the paper introduces two new defined objects (fineness index and neat trees) whose independent support is not visible. No free parameters or explicit axioms are stated in the provided text.

invented entities (2)
  • fineness index fi(P) no independent evidence
    purpose: Captures the threshold where a poset evades domination by countable subsets, generalizing b
    New cardinal invariant introduced to establish the universal lower bound on character.
  • neat trees no independent evidence
    purpose: Refine uniform covering trees to prove strict Tukey equivalence
    Novel machinery developed to overcome structural obstructions in lifting character equalities.

pith-pipeline@v0.9.1-grok · 5778 in / 1337 out tokens · 29995 ms · 2026-06-29T19:53:24.306836+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

27 extracted references

  1. [1]

    Arhangel’skii and M.G

    A.V. Arhangel’skii and M.G. Tkachenko,Topological Groups and Related Structures, Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008

    2008

  2. [2]

    Banakh, Topological spaces with anωω-base,Dissertationes Math.538(2019), 141 pp

    T. Banakh, Topological spaces with anωω-base,Dissertationes Math.538(2019), 141 pp

    2019

  3. [3]

    Banakh and A

    T. Banakh and A. Leiderman,ωω-dominated function spaces andωω-bases in free objects of topological algebra,Topology Appl.241(2018), 203–241

    2018

  4. [4]

    Cascales and J

    B. Cascales and J. Orihuela, On compactness in locally convex spaces,Math. Z.195(1987), 365–381

    1987

  5. [5]

    Chis, M.V

    C. Chis, M.V. Ferrer, S. Hernández and B. Tsaban, Boaz, The character of topological groups, via bounded systems, Pontryagin–van Kampen duality and pcf theory,J. Algebra420(2014), 86–119

    2014

  6. [6]

    Dow and Z

    A. Dow and Z. Feng, Compact spaces with aP-base,Indag. Math. (N.S.)32(2021), no. 4, 777–791

    2021

  7. [7]

    Engelking,General Topology, 2nd ed., Sigma Series in Pure Mathematics, vol

    R. Engelking,General Topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989

    1989

  8. [8]

    Feng,P-bases and topological groups,Proc

    Z. Feng,P-bases and topological groups,Proc. Amer. Math. Soc.150(2022), no. 2, 877–889

    2022

  9. [9]

    Feng and P

    Z. Feng and P. Gartside. Directed sets of topology: Tukey representation and rejection,Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.118(2024), 44

    2024

  10. [10]

    Gabriyelyan, J

    S. Gabriyelyan, J. Kąkol and A. Leiderman, On topological groups with a small base and metrizability, Fund. Math.229(2015), 129–158

    2015

  11. [11]

    Gartside, Tukey order and diversity of free Abelian topological groups,J

    P. Gartside, Tukey order and diversity of free Abelian topological groups,J. Pure Appl. Algebra225(2021), 106712. 16 X. GONG AND D. PENG

    2021

  12. [12]

    Gartsice and A

    P. Gartsice and A. Mamatelashvili, The Tukey order on compact subsets of separable metric spaces,J. Symb. Logic81(2016), no.1, 181–200

    2016

  13. [13]

    W. He, D. Peng, M. Tkachenko and H. Zhang,M-factorizability of products andτ-fine topological groups, Topology Appl.296(2021), 107674

    2021

  14. [14]

    Huang, OnP-bases andωω-sn-networks,Topology Appl.341(2024), 108758

    Y. Huang, OnP-bases andωω-sn-networks,Topology Appl.341(2024), 108758

    2024

  15. [15]

    Isbell, Seven cofinal types,J

    J. Isbell, Seven cofinal types,J. London Math. Soc.4(1972), 651–654

    1972

  16. [16]

    Kuzeljevića and S

    B. Kuzeljevića and S. Milošević, Cofinal types and topological groups,Topology Appl.356(2024), 109051

    2024

  17. [17]

    Leiderman, V

    A. Leiderman, V. Pestov and A. Tomita, On topological groups admitting a base at the identity indexed byωω,Fund. Math.238(2017), 79–100

    2017

  18. [18]

    F. Lin, A. Ravsky and J. Zhang, Countable tightness andG-bases on free topological groups,Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.114(2020), 67

    2020

  19. [19]

    Lipin, E

    A. Lipin, E. Reznichenko and O. Sipacheva,Rω1-factorizable spaces and groups,Topology Appl.382(2026), 109760

    2026

  20. [20]

    Mamatelashvili, Tukey order on sets of compact subsets of topological spaces, Thesis (Ph.D.)–University of Pittsburgh

    A. Mamatelashvili, Tukey order on sets of compact subsets of topological spaces, Thesis (Ph.D.)–University of Pittsburgh. ProQuest LLC, Ann Arbor, MI, 2014. 141 pp

    2014

  21. [21]

    Nickolas and M

    P. Nickolas and M. Tkachenko, The character of free topological groups. I.Appl. Gen. Topol.6(2005), no. 1, 15–41

    2005

  22. [22]

    Nickolas and M

    P. Nickolas and M. Tkachenko, The character of free topological groups. II.Appl. Gen. Topol.6(2005), no. 1, 43–56

    2005

  23. [23]

    Schmidt, Konfinalität,Z

    J. Schmidt, Konfinalität,Z. Math. Logik Grundlagen Math.1(1955), 271–303

    1955

  24. [24]

    Shelah, Advances in cardinal arithmetic

    S. Shelah, Advances in cardinal arithmetic. Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), 355–383

    1991

  25. [25]

    Todorčević, Directed sets and cofinal types.Trans

    S. Todorčević, Directed sets and cofinal types.Trans. Amer. Math. Soc.290(1985), no. 2, 711–723

    1985

  26. [26]

    Tukey, Convergence and uniformlty in topology,.Ann

    J. Tukey, Convergence and uniformlty in topology,.Ann. of Math. Studiesno. 2, Princeton Univ. Press, Princeton, N. J., 1940

    1940

  27. [27]

    Zhang, W

    H. Zhang, W. He and W. Xi, The fineness index of topological groups and M-factorizable groups,Topology Appl.328(2023), 108463. (X. Gong)Institute of Mathematics, Nanjing Normal University, Nanjing 210024, China Email address:gongxuan2021@163.com (D. Peng)Institute of Mathematics, Nanjing Normal University, Nanjing 210024, China Email address:pengdk10@lzu.edu.cn

    2023