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arxiv: 2508.13443 · v2 · submitted 2025-08-19 · 🧮 math.GN · math.GR

Suitable sets for topological groups revisited

Fucai Lin , Jiamin He , Jiajia Yang , Chuan Liu This is my paper

Pith reviewed 2026-05-18 23:05 UTC · model grok-4.3

classification 🧮 math.GN math.GR
keywords suitable setstopological groupsω^ω-basek-spacesmetrizabilityfree topological groupslinear orderabilitycardinal invariants
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The pith

Linearly orderable topological groups with an ω^ω-base are metrizable and have suitable sets

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

The paper proves that every linearly orderable topological group with an ω^ω-base is metrizable, which implies it has a suitable set. It also shows that any k-space topological group with an ω^ω-base has a suitable set. For non-separable k_ω-spaces without non-trivial convergent sequences, the free topological group A(X) being snf-countable means it lacks a suitable set. These results provide partial answers to open problems about the existence of suitable sets in topological groups.

Core claim

In this paper, we prove that every linearly orderable topological group with an ω^ω-base is metrizable and thus possesses a suitable set. Additionally, we show that each topological group G with an ω^ω-base has a suitable set provided that G is a k-space. We also establish a non-existence result for suitable sets in certain free topological groups A(X) where X is a non-separable k_ω-space without non-trivial convergent sequences when A(X) is snf-countable.

What carries the argument

Suitable sets, which are discrete subsets S of G such that S ∪ {e} is closed and the subgroup generated by S is dense in G, along with the ω^ω-base property used to establish metrizability under linear orderability.

If this is right

  • Linearly orderable groups with ω^ω-bases are metrizable.
  • Topological groups with ω^ω-bases that are k-spaces have suitable sets.
  • Cardinal invariants can be studied for groups possessing suitable sets.
  • Non-existence of suitable sets holds in some free topological groups under the given conditions.

Where Pith is reading between the lines

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

  • The ω^ω-base imposes strong structural restrictions when paired with linear orderability.
  • Suitable sets may serve as a tool to study density and generation properties across broader classes of topological groups.
  • The non-existence result highlights conditions where free topological groups cannot be generated densely by discrete closed sets.

Load-bearing premise

The group is linearly orderable to conclude metrizability from the presence of an ω^ω-base, or X is a non-separable k_ω-space without non-trivial convergent sequences for the non-existence result in A(X).

What would settle it

A concrete counterexample would be a linearly orderable topological group with an ω^ω-base that fails to be metrizable.

read the original abstract

A discrete subset $S$ of a topological group $G$ is called a {\it suitable set} for $G$ if $S\cup \{e\}$ is closed in $G$ and the subgroup generated by $S$ is dense in $G$, where $e$ is the identity element of $G$. In this paper, the existence of suitable sets in topological groups is studied. It is proved that, for a non-separable $k_{\omega}$-space $X$ without non-trivial convergent sequences, the $snf$-countability of $A(X)$ implies that $A(X)$ does not have a suitable set, which gives a partial answer to \cite[Problem 2.1]{TKA1997}. Moreover, the existence of suitable sets in some particular classes of linearly orderable topological groups is considered, where Theorem~\ref{t4} provides an affirmative answer to \cite[Problem 4.3]{ST2002}. Then, topological groups with an $\omega^{\omega}$-base are discussed, and every linearly orderable topological group with an $\omega^{\omega}$-base being metrizable is proved; thus it has a suitable set. Further, it follows that each topological group $G$ with an $\omega^{\omega}$-base has a suitable set whenever $G$ is a $k$-space, which gives a generalization of a well-known result in \cite{CM}. Finally, some cardinal invariant of topological groups with a suitable set are provided. Some results of this paper give some partial answers to some open problems posed in~\cite{DTA} and~\cite{TKA1997} respectively.

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

0 major / 3 minor

Summary. The paper studies suitable sets in topological groups, where a discrete subset S is suitable if S union {e} is closed and the subgroup generated by S is dense. It proves that for a non-separable k_ω-space X without non-trivial convergent sequences, snf-countability of the free topological group A(X) implies A(X) has no suitable set, partially answering Problem 2.1 from TKA1997. It establishes existence results for suitable sets in certain linearly orderable topological groups, with Theorem 4 affirmatively answering Problem 4.3 from ST2002. It proves every linearly orderable topological group with an ω^ω-base is metrizable (hence has a suitable set) and that any topological group with an ω^ω-base that is a k-space has a suitable set, generalizing a result from CM. The paper also provides some cardinal invariants for topological groups admitting suitable sets.

Significance. If the central claims hold, the work provides targeted partial resolutions to open problems on suitable sets and introduces a metrizability criterion for linearly orderable groups under the ω^ω-base condition, which directly yields existence of suitable sets. The extension to k-spaces broadens an earlier result from CM without apparent circularity. The cardinal invariants add quantitative structure. These results are of interest in the study of topological groups and free topological groups.

minor comments (3)
  1. [§3] §3 (proof of the metrizability result for linearly orderable groups): the reduction of character to ℵ₀ via order compatibility and the directed ω^ω-base at the identity is central; an explicit intermediate step showing how the base elements are nested under the order would improve readability.
  2. [Introduction] The citation [CM] is used for the known result being generalized; ensure the bibliography entry is complete and the precise statement being generalized is quoted or restated in the introduction.
  3. [§2] In the non-existence result for A(X), the hypotheses (non-separable k_ω-space, no non-trivial convergent sequences, snf-countable) are correctly conditional; verify that the definition of snf-countability is recalled or referenced at first use in §2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The provided summary accurately captures the main results on suitable sets in topological groups, including the partial answer to Problem 2.1 from TKA1997 and the affirmative answer to Problem 4.3 from ST2002, as well as the metrizability criterion and generalization of the result from CM.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper's central results—the metrizability of linearly orderable groups with an ω^ω-base via order compatibility and directed base at the identity, plus the existence of suitable sets for k-spaces with such bases—rely on new proofs that interact standard topological properties without reducing to the paper's own fitted parameters, self-definitions, or load-bearing self-citations. The non-existence claim for A(X) is explicitly conditional on external hypotheses (non-separable k_ω-space without convergent sequences and snf-countability). Citations to prior works such as [CM] supply independent known results rather than unverified self-referential chains, and no ansatz, renaming, or uniqueness theorem is smuggled in via author-overlapping references. The derivation chain remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on established axioms of topology and group theory without introducing free parameters or new entities.

axioms (1)
  • standard math Standard axioms of topological spaces, groups, and free topological groups
    Invoked in all definitions of suitable sets, kω-spaces, and metrizability arguments.

pith-pipeline@v0.9.0 · 5834 in / 1140 out tokens · 56023 ms · 2026-05-18T23:05:26.191943+00:00 · methodology

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Arhangel’shiˇ ı, M

    A.V. Arhangel’shiˇ ı, M. Tkachenko,Topological Groups and Related Structures, Atlantis Press and World Sci., 2008

    work page 2008

  2. [2]

    Banakh, Topological spaces with an ωω, Diss

    T. Banakh, Topological spaces with an ωω, Diss. Math. 538(2019) 1-141

    work page 2019

  3. [3]

    Comfort, S

    W.W. Comfort, S. Garc´ ıa-Ferreira, Resolvability: a selective survey and some new results , Topol. Appl., 1996, 74: 149–167

    work page 1996

  4. [4]

    Comfort, S.A

    W.W. Comfort, S.A. Morris, D. Robbie, et al., Suitable sets for topological groups , Topol. Appl., 1998, 86(1): 25–46. 14 FUCAI LIN*, JIAMIN HE, JIAJIA YANG, AND CHUAN LIU

    work page 1998

  5. [5]

    Dikranjan, M

    D. Dikranjan, M. Tkacenko, V. Tkachuk, Some topological groups with and some without suitable sets, Topol. Appl., 1999, 98(1-3): 131–148

    work page 1999

  6. [6]

    Dikranjan, M

    D. Dikranjan, M. Tkaˇ cenko, V. Tkachuk, Topological groups with thin generating sets , J. Pure Appl. Algebra, 2000, 145(2): 123–148

    work page 2000

  7. [7]

    Dikranjan, F

    D. Dikranjan, F. Trigos-Arrieta, Suitable sets in some topological groups , J. Group Theorey, 2000, 3(1-3): 293–321

    work page 2000

  8. [8]

    Douady, Cohomologie des groupes compacts totalement discontinus , in S´ eminaire Bour- baki, 12e ann´ ee, 1959/60, second edition,189, Secr´ etariat Math., Paris, 1960, 287–298

    A. Douady, Cohomologie des groupes compacts totalement discontinus , in S´ eminaire Bour- baki, 12e ann´ ee, 1959/60, second edition,189, Secr´ etariat Math., Paris, 1960, 287–298

    work page 1959

  9. [9]

    Gabriyelyan, J

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

    work page 2015

  10. [10]

    Gruenhage, Generalized metric spaces, Handbook of Set-Theoretic Topology, North- Holland, Amsterdam, 1984, 423-501

    G. Gruenhage, Generalized metric spaces, Handbook of Set-Theoretic Topology, North- Holland, Amsterdam, 1984, 423-501

    work page 1984

  11. [11]

    Hofmann, S.A

    K.H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Alge., 1990, 68: 181–194

    work page 1990

  12. [12]

    F. Lin, C. Liu C, J. Cao, Two weak forms of countability axioms in free topological groups, Topol. Appl., 2016, 207: 96–108

    work page 2016

  13. [13]

    F. Lin, C. Liu, The characterizations of hyperspaces and free topological groups with an ωω-base, arXiv:2502.19727

    work page arXiv

  14. [14]

    F. Lin, A. Ravsky, J. Zhang, Countable tightness and G-bases on free topological groups, RACSAM, 2020, 114: 67

    work page 2020

  15. [15]

    Lin, On sequence-covering s-maps, Adv

    S. Lin, On sequence-covering s-maps, Adv. in Math. (China), 25 (1996), 548–551

    work page 1996

  16. [16]

    Malykhin, Extremally disconnected topological groups, Uspekhin Mat

    V.I. Malykhin, Extremally disconnected topological groups, Uspekhin Mat. Nauk, 1979, 34(6): 59–66

    work page 1979

  17. [17]

    Mel’nikov, Normal divisors of free profinite groups, Math

    O.V. Mel’nikov, Normal divisors of free profinite groups, Math. USSR-Izv. 1978, 12(1): 1–20

    work page 1978

  18. [18]

    Migacz, Convergent sequences in various topological spaces, arXiv:2107.04625

    D. Migacz, Convergent sequences in various topological spaces, arXiv:2107.04625

    work page arXiv

  19. [19]

    Nyikos, H.C

    P.J. Nyikos, H.C. Reichel, Topologically orderable groups , Gen. Topol. Appl., 1975, 5(3): 195–204

    work page 1975

  20. [20]

    Nyikos, H

    P.J. Nyikos, H. Reichel, Topologically orderable groups, Gen. Topol. Appl.,11(1980), 281–292

    work page 1980

  21. [21]

    Sanchis, M

    M. Sanchis, M. Tkachenko, Suitable sets for subgroups of direct sums of discrete groups , J. Pure Appl. Alge., 171(2002), 513–315–332

    work page 2002

  22. [22]

    Tkachenko, Generating dense subgroups of topological groups, Topol

    M. Tkachenko, Generating dense subgroups of topological groups, Topol. Proc. 1997, 22: 533– 582

    work page 1997

  23. [23]

    Venkataraman, M

    M. Venkataraman, M. Rajagopalan, T. Soundararajan, Orderable topological spaces, Gen. Topol. Appl., 1972, 2(1): 1–10. (Fucai Lin): Department of Mathematics and Information Science, Zhangzhou Nor- mal University, Zhangzhou 363000, P. R. China Email address: linfucai2008@aliyun.com; linfucai@mnnu.edu.cn (Jiamin He): School of mathematics and statistics, Mi...

    work page 1972