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REVIEW 2 major objections 3 minor 27 references

On convergent sequences in dual groups

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a precompact torsion abelian group, a nontrivial convergent sequence exists in the dual group exactly when the group has a countably infinite Hausdorff quotient.

desk verdict Torsion characterization and a counterexample to the Baire-property conjecture are solid and new; one proof step leans on an unstated duality theorem and the abstract has a typo, but both are fixable. read the letter →

arxiv 1908.03415 v2 pith:IENCAN3E submitted 2019-08-09 math.GN math.FA

M.V. Ferrer , S. Hernández , M. Tkachenko This is my paper · ORCID
classification math.GNmath.FA MSC 43A4022D3522C0554E5254C10
keywords reflexiveprecompactpseudocompactBairepropertyconvergentsequencedualgrouptorsionabelianPontryaginduality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a duality-theoretic characterization for precompact abelian torsion groups $G$ (abelian groups that sit densely inside a compact group and whose elements all have finite order): the dual group $G_p^\wedge$---the continuous characters of $G$ with the topology of pointwise convergence---contains a nontrivial convergent sequence exactly when $G$ has a countably infinite Hausdorff quotient. This reduces a subtle analytic property of the dual to a purely algebraic property of the original group. Along the way the authors show that this is equivalent to the dual containing an infinite metrizable subgroup and to the group admitting a countably infinite Hausdorff homomorphic image. They also construct a dense subgroup of $\mathbb{Z}(2)^\omega$ that is first category and has measure zero, yet whose dual has no nontrivial convergent sequences, and they use this to exhibit a precompact reflexive abelian group of the first Baire category.

What carries the argument

The central object is the dual group $G_p^\wedge$: all continuous homomorphisms from $G$ to the circle group $\mathbb{T}$, with the topology of pointwise convergence on elements of $G$. Two mechanisms carry the proof. First, the canonical evaluation mapping $G\to(G_p^\wedge)_p^\wedge$ is taken to be a topological isomorphism for precompact abelian $G$, so a closed metrizable subgroup $\Gamma$ of $G_p^\wedge$ can be converted into the quotient $G/\Gamma^\perp$. Second, for a convergent sequence of characters, the diagonal product $f=\triangle_n\chi_n:G\to\mathbb{T}^\omega$ has image in the direct sum $D^{(\omega)}$ when $G$ is torsion, which makes the image countable and yields the quotient witnessing the sequence. The paper also uses the group $c_0(\mathbb{T})$ of sequences in $\mathbb{T}$ converging to $1$: a dual group contains a nontrivial convergent sequence exactly when $G$ maps onto a subgroup of $c_0(\mathbb{T})$ that separates the points of the underlying convergent sequence.

What would settle it

A single counterexample would settle it: a precompact torsion abelian group $G$ such that $G_p^\wedge$ has a nontrivial sequence converging to the identity while every infinite Hausdorff quotient of $G$ is uncountable. The paper's diagonal-product proof predicts such a sequence would produce a countably infinite quotient, so no such group exists.

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Extended reading notes

Core claim

The main theorem (Theorem 2.7) states that for a precompact torsion abelian group $G$, the following are equivalent: (1) $G$ has a countably infinite Hausdorff quotient; (2) $G$ has a countably infinite Hausdorff homomorphic image; (3) $G_p^\wedge$ contains an infinite metrizable subgroup; and (4) $G_p^\wedge$ contains a nontrivial convergent sequence. The proof of (4) implies (2) is the heart: a nontrivial sequence of characters converging to the identity gives a diagonal homomorphism $f:G\to\mathbb{T}^\omega$, and because $G$ is torsion each value $f(x)$ has finite order, so $f(x)$ lies in the direct-sum subgroup $D^{(\omega)}$ of $\mathbb{T}^\omega$, where $D$ is the torsion subgroup of the circle group; hence $f(G)$ is a countably infinite Hausdorff homomorphic image of $G$. The converse direction is also explicit: the dual of a countably infinite precompact group is an infinite metrizable group, which embeds in $G_p^\wedge$ and therefore supplies a convergent sequence.

Load-bearing premise

The proof leans on the cited theorem that every precompact abelian group is topologically isomorphic to its second dual; if that theorem only gives a dense embedding in general, the step converting a closed metrizable subgroup of the dual into a quotient of $G$ breaks.

Editorial extensions

If this is right

  • For precompact torsion groups, the search for convergent sequences in the dual is a search for countable quotients: if every infinite quotient of $G$ is uncountable, then $G_p^\wedge$ has no nontrivial convergent sequence.
  • Every nontrivial convergent sequence in $G_p^\wedge$ factors through a countably infinite Hausdorff quotient of $G$; the diagonal homomorphism built from the sequence has that quotient as its image.
  • Countably infinite precompact torsion groups are the extreme case: their duals are infinite metrizable groups and hence contain convergent sequences.
  • There exist precompact abelian groups whose duals contain infinite compact subsets but no nontrivial convergent sequences, so the torsion characterization does not extend verbatim to all precompact groups.
  • The first-category, measure-zero subgroup of $\mathbb{Z}(2)^\omega$ whose dual has only finite bounded subsets yields a precompact reflexive abelian group of the first Baire category, showing that reflexivity for precompact groups does not force the Baire property.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension question, not answered in the paper, is whether absence of nontrivial convergent sequences in the dual of a precompact torsion group forces every compact subset of the dual to be finite; the paper proves this for its example and cites it for bounded torsion groups, but not for all torsion groups.
  • The torsion hypothesis enters only to make the diagonal image $f(G)$ countable; outside the torsion setting the same argument can produce uncountable images, which suggests the equivalence between convergent sequences and countable quotients is genuinely special to torsion groups.
  • Because the proof of (3) implies (2) depends on the cited double-duality theorem, a reader who wants to rely on Theorem 2.7 should verify that theorem's exact hypotheses; if it only gives a dense embedding for some precompact groups, the annihilator step would need repair before the equivalence is established.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper studies precompact abelian groups whose dual group, equipped with the pointwise convergence topology, contains a nontrivial convergent sequence. The main theorem (Theorem 2.7) states that for a precompact torsion abelian group, four conditions are equivalent: (1) G has a countably infinite Hausdorff quotient, (2) G has a countably infinite Hausdorff homomorphic image, (3) the dual group G_p^ contains an infinite metrizable subgroup, and (4) G_p^ contains a nontrivial convergent sequence. The paper also constructs, in Theorem 2.3, a dense subgroup of Z(2)^ω that is of the first category, has measure zero, and whose dual contains no infinite compact subsets; as a consequence it obtains a first-category precompact reflexive abelian group in Theorem 3.9. Section 3 develops a general criterion for the presence of compact subsets in dual groups in terms of separating subgroups of Cp(K, T).

Significance. If the main results hold, Theorem 2.7 gives a clean internal characterization of convergent sequences in duals of torsion precompact abelian groups, reducing the phenomenon to a countable quotient property of the original group. The construction in Theorem 2.3 and the reflexive first-category group in Theorem 3.9 are interesting and complement the earlier work of Hart and Kunen. The paper is largely self-contained in its combinatorial construction, uses standard duality tools, and contains no fitted parameters or circular arguments. The proofs are mostly direct, and the overall contribution is a useful step in understanding compactness and convergence in duals of precompact groups.

major comments (2)
  1. [Abstract and Introduction] The abstract and the introduction both state that the torsion characterization is 'No infinite quotient group of G is countable.' This is the negation of condition (1) of Theorem 2.7, which asserts that the existence of a nontrivial convergent sequence is equivalent to G having a countably infinite Hausdorff quotient. As written, the advertised characterization is the exact opposite of the theorem proved in the paper. This is a load-bearing misstatement of the main result and must be corrected in both places.
  2. [Proposition 2.5, proof of (3)⇒(2), and Theorem 3.9] The proof of Proposition 2.5(3)⇒(2) invokes the unstated theorem from [25] that 'the canonical evaluation mapping of G to (G_p^)_p^ is a topological isomorphism.' The same external result is used again in the proof of Theorem 3.9. The authors neither quote the theorem nor verify its hypotheses for the groups in question. This is a load-bearing step because it is used to identify G with its double dual and then to conclude G/Γ^⊥ ≅ Γ^. If [25] provides only a topological embedding or requires additional hypotheses, the argument fails as written. In fact, a direct proof of (3)⇒(2) is available without any biduality: for Γ an infinite metrizable subgroup of G_p^, the dual Γ^ is countable because Γ is precompact metrizable, and the evaluation map Φ: G → Γ^ has kernel Γ^⊥ (closed) and image isomorphic to G/Γ^⊥; the image is infinite because otherwise Γ would embed into the finite dual of the finite quotient G/Γ^⊥, forcing Γ to be finite. The authors should either state the relevant theorem from [25] and confirm it applies to all precompact abelian groups, or replace the argument with a direct one.
minor comments (3)
  1. [Theorem 2.3, proof] In the final paragraph of the proof, the sentence 'This implies that G does not contain non-trivial convergent sequences at all' should refer to the dual group G_p^, not to G. The proof has just shown that no sequence in G_p^ converges to the identity.
  2. [Proposition 2.5, proof of (2)⇒(1)] There is a formatting typo in the first line: 'Let φ : G − → H' should read 'Let φ : G → H'. This does not affect the mathematics.
  3. [References] Reference [25] is cited for a central duality result without a theorem number or page reference. Since this citation is used in two load-bearing proofs, a more precise pointer would help the reader verify the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main equivalences are proved from external duality theorems and direct torsion-group computations.

full rationale

The derivation chain in Proposition 2.5 and Theorem 2.7 does not reduce to its own inputs. In Proposition 2.5, (1) implies (3) uses the standard embedding of the dual of a quotient into G_p^; (3) implies (2) invokes the biduality theorem of Raczkowski and Trigos-Arrieta [25] to identify a precompact abelian group G with (G_p^)_p^ and turn a closed metrizable subgroup Gamma of G_p^ into a countably infinite Hausdorff quotient G/Gamma^perp. This is an external, independently published theorem, not a reformulation of the paper's target, and the step is not circular even though the theorem is cited rather than restated. Theorem 2.7's only non-formal implication, (4) implies (2), is a direct computation: for a torsion group each chi_n(x) lies in a finite cyclic subgroup of T, so convergence to 1 forces eventual equality to 1; hence the diagonal map f has countable image, and the infinite family of induced characters shows the image is infinite. Theorem 2.3 is an explicit construction independent of the characterization. Self-citations (e.g. [2], [4], [19]) are used as background results for h-embedded subgroups, precompact reflexive groups, and dual boundedness; they are not cited to establish the equivalence that is the paper's main claim. No fitted parameter is relabelled as a prediction, no quantity is defined in terms of what it is used to prove, and no uniqueness claim is imported from the authors' own prior work. The only substantive external-dependency concern is whether [25] really supplies a topological isomorphism for all precompact abelian groups; that is a correctness check, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters and no newly postulated entities. The paper constructs explicit subgroups and imports standard or prior theorems. The only non-standard import is the biduality theorem [25], which is load-bearing and not stated.

assumptions (6)
  • domain assumption Every precompact abelian group is determined by its continuous characters, and its topology is the topology of pointwise convergence on the dual (Comfort-Ross, [6, Theorem 1.2]).
    Used throughout to identify G with characters and to justify the pointwise convergence topology on the dual.
  • standard math Every continuous character of Z(2)^omega depends on finitely many coordinates (Kaplan, [21]).
    Used in Theorem 2.3 to reduce arbitrary sequences in the dual to finite support sets.
  • domain assumption The canonical evaluation mapping G to (G_p^)_p^ is a topological isomorphism for precompact abelian G ([25]).
    Load-bearing in Proposition 2.5(3) implies (2); the paper does not state the theorem or verify its hypotheses.
  • standard math If a compact space is countably infinite, it contains a nontrivial convergent sequence.
    Used in Theorem 2.3 to conclude all compact subsets of the countable dual are finite once bounded subsets are finite.
  • standard math For a topological group, homogeneous spaces that are not Baire are of the first category in themselves (Lutzer-McCoy, [22, Theorem 2.3]).
    Used in the introduction and for the first-category statements.
  • domain assumption [4, Theorem 4.2] provides a pseudocompact group S with a closed pseudocompact subgroup P such that S/P is isomorphic to Z(2)^omega and all countable subgroups of S are h-embedded.
    Used in Theorem 3.9 to build the reflexive example; imported from cited prior work.

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Pith. "Pith review of On convergent sequences in dual groups." pith.science (2026). https://pith.science/paper/IENCAN3E

@misc{pith2026190803415,
  author       = {Pith},
  title        = {Pith review of: On convergent sequences in dual groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IENCAN3E}},
  note         = {Machine review of arXiv:1908.03415}
}
abstract

We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact abelian \emph{torsion} groups $G$, we characterize the existence of a nontrivial convergent sequence in $G_p^\wedge$ by the following property of $G$: \emph{No infinite quotient group of $G$ is countable.} Finally, we present an example of a dense subgroup $G$ of the compact metrizable group $\mathbb{Z}(2)^\omega$ such that $G$ is of the first category in itself, has measure zero, but the dual group $G_p^\wedge$ does not contain infinite compact subsets. This complements Theorem 1.6 in [J.E.~Hart and K.~Kunen, Limits in function spaces and compact groups, \textit{Topol. Appl.} \textbf{151} (2005), 157--168]. As a consequence, we obtain an example of a precompact reflexive abelian group which is of the first Baire category.

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