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Simply $sm$-factorizable (para)topological groups and their completions

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Pith's one-line read A group is simply sm-factorizable exactly when every real-valued function factors through a strongly submetrizable group, and then all three major completions agree.

desk verdict Solid paper—the main theorem answers a real open question and the core proof is coherent; worth refereeing. read the letter →

arxiv 1908.08627 v2 pith:2OQD262D submitted 2019-08-23 math.GN

Li-Hong Xie , Mikhail Tkachenko This is my paper · ORCID
classification math.GN MSC 22A0522A3054H1154A2554C30
keywords simplysm-factorizableparatopologicalgrouptopologicalstronglysubmetrizablerealcompactificationDieudonnécompletionR-factorizableRaikov
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 pins down what it means for a (para)topological group to be simply sm-factorizable: every continuous real-valued function on the group must factor through a continuous homomorphism onto a strongly submetrizable group, meaning a group with a coarser separable metrizable group topology. Using that characterization, the paper proves that for every Hausdorff simply sm-factorizable topological group, the Dieudonné completion $\mu G$, the Hewitt-Nachbin realcompactification $\upsilon G$, and the $G_\delta$-closure $\varrho_\omega G$ of the Raikov completion are one and the same object. That identity makes the group completion friendly, so its Dieudonné completion carries a topological group structure extending the original operations. For regular simply sm-factorizable paratopological groups, the realcompactification is itself a simply sm-factorizable paratopological group containing the original group as a dense subgroup, and it coincides with the Dieudonné completion. These results answer a 2018 open question and improve earlier completion theorems for paratopological groups.

What carries the argument

The load-bearing object is the strongly submetrizable group: a (para)topological group admitting a coarser separable metrizable (para)topological group topology, equivalently a continuous one-to-one homomorphism onto a separable metrizable group. Theorem 2.4 upgrades the definition of simple sm-factorizability to a factorization through these groups. Because strongly submetrizable groups have countable pseudocharacter and are Moscow (closures of open sets are unions of $G_\delta$-sets), continuous functions on them extend across the $G_\delta$-closure of their Raikov completion; Proposition 3.1 uses this to show that a $G_\delta$-dense simply sm-factorizable subgroup is $C$-embedded in its ambient group. Theorem 3.2 then combines this $C$-embeddedness with the bound $c(G) \leq 2^{ib(G)} \leq 2^{\mathfrak c}$ and the Ulam non-measurability of $2^{\mathfrak c}$ to force $\mu G = \upsilon G$. For paratopological groups, the corresponding mechanism is the diagonal embedding of $G$ into a product of strongly submetrizable quotients; Lemma 4.2, via closure of countable Hausdorff number under products and subgroups, makes the $G_\delta$-closure of that diagonal a subgroup, so the realcompactification inherits the group structure.

What would settle it

Find a $G_\delta$-dense simply sm-factorizable subgroup $H$ of a topological group $G$ and a continuous real-valued function on $H$ that has no continuous extension over $G$; Proposition 3.1 would be false, and the equality $\mu G = \varrho_\omega G = \upsilon G$ for the larger group would no longer be derivable.

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Extended reading notes

Core claim

The central claim is Theorem 3.2: if $G$ is a Hausdorff simply sm-factorizable topological group, then $\mu G = \varrho_\omega G = \upsilon G$, so $G$ is completion friendly and a $PT$-group. The engine is Theorem 2.4, which makes an apparently weaker defining property equivalent to a strong one: $G$ is simply sm-factorizable if and only if every continuous $f\colon G \to \mathbb{R}$ is $f = g \circ \pi$ for a continuous homomorphism $\pi$ onto a strongly submetrizable (regular) group and a continuous $g$. On the paratopological side, Theorem 4.3 says the realcompactification of a regular simply sm-factorizable paratopological group carries a natural paratopological group structure, contains $G$ as a dense subgroup, and is itself simply sm-factorizable; Corollary 4.5 adds $\mu G = \upsilon G$, and Theorem 4.20 gives the same conclusion when the associated topological group is $\omega$-narrow and simply sm-factorizable.

Load-bearing premise

The paratopological completion theorems stand on the cited preservation fact that paratopological groups with countable Hausdorff number are closed under arbitrary products and subgroups; if that fact failed, the $G_\delta$-closure of the diagonal embedding could fail to be a subgroup and the realcompactification would not inherit the group structure.

Editorial extensions

If this is right

  • Every Hausdorff simply sm-factorizable topological group is a PT-group: its Dieudonné completion is a topological group containing G as a dense subgroup (Corollary 3.3).
  • Every Hausdorff weakly Lindelöf topological group satisfies μG = ρωG = υG and is completion friendly (Corollary 3.4).
  • For every regular simply sm-factorizable paratopological group, μG = υG, and that common completion is a simply sm-factorizable paratopological group containing G densely (Corollary 4.5).
  • If the topological group G* associated to a regular paratopological group G is ω-narrow and simply sm-factorizable, then G is simply sm-factorizable and υG carries a paratopological group structure (Theorems 4.13 and 4.20).
  • If G* is R-factorizable rather than merely simply sm-factorizable, then the same realcompactification conclusions hold and μG = υG (Corollaries 4.18 and 4.21).

Reading between the lines

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

  • The factorization characterization suggests a practical test for simple sm-factorizability: for any group with an explicit presentation, check whether every continuous real-valued function descends to a strongly submetrizable quotient; this may be easier than checking co-zero sets directly.
  • A natural open direction, not pursued in the paper, is whether the equality μG = ρωG = υG is preserved by dense Gδ-subgroups or countable products of Hausdorff simply sm-factorizable groups; the proof route through C-embeddedness does not automatically close under these operations.
  • For paratopological groups, the key role of countable Hausdorff number indicates where to look for a counterexample: a regular simply sm-factorizable paratopological group whose realcompactification is not a Gδ-closure of a diagonal embedding would separate the construction from the conclusion.
  • Example 4.17 shows the associated topological group, not G itself, controls the relevant narrowness; one testable extension is to characterize paratopological groups G for which G* is simply sm-factorizable in terms of factorization through G*.
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Editorial analysis

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

Referee Report

0 major / 7 minor

Summary. The paper introduces a class of (para)topological groups called strongly submetrizable and characterizes simply sm-factorizable groups by the property that every continuous real-valued function factors through a continuous homomorphism onto a strongly submetrizable (para)topological group (Theorem 2.4). It also gives an ω-narrow version of this characterization using invariant admissible subgroups (Theorem 2.15). The main result for topological groups is Theorem 3.2, which states that for every Hausdorff simply sm-factorizable topological group G one has μG = ρωG = υG, so G is completion friendly and hence a PT-group; this answers Problem 1.2. For paratopological groups, Section 4 proves that regular simply sm-factorizable groups satisfy μG = υG and that this completion carries a paratopological group structure and remains simply sm-factorizable, with several related results and examples.

Significance. Theorem 3.2 resolves an open problem of Arhangel'skii and Tkachenko and unifies several earlier completion results for weakly Lindelöf and R-factorizable groups. The characterization in Theorem 2.4 is clean and likely to be a useful tool for further work on factorization properties. The proof of the main theorem is coherent and self-contained up to standard cited facts about Moscow spaces, Gδ-density, and Ulam non-measurable cardinals. The examples and boundary results in Section 4, especially Example 4.17, help delineate the scope of the theory, which is a strength of the paper.

minor comments (7)
  1. [§2, proof of Theorem 2.4] In the proof of (1) ⇒ (2), the symbol h appears as a map with domain G before it is introduced; the reader must guess that h is the diagonal homomorphism π. Please define the quotient map and the space H explicitly, and avoid using h for two different purposes.
  2. [§3, Theorem 3.2 and surrounding text] The sentence 'The following result gives a positive answer to Problem 1.3' before Theorem 3.2 is inconsistent with the numbering in the introduction, where Theorem 3.2 is said to answer Problem 1.2. Problem 1.3 concerns continuous homomorphic images, not completions; please correct the cross-reference.
  3. [§4, proof of Theorem 4.3] In the sentence 'the realcompactification υG of G admits a natural structure of a topological group containing G as a dense subgroup', the phrase 'topological group' should be 'paratopological group', since the construction yields a paratopological group structure.
  4. [§4, proof of Theorem 4.13] The notation Ir(G) is used without definition; if it denotes the index of narrowness, please replace it with ib(G), which is defined in the introduction, and ensure consistency with the cited result [14, Theorem 2].
  5. [§4, Example 4.17] The equality ⟨U_B⟩ = {x ∈ G : x(α) = 0 for each α ∈ B} is asserted with the phrase 'An easy verification shows'. Since the subgroup S constructed in Lemma 4.16 is not evidently closed under coordinatewise positive/negative parts, this equality is not immediate; please supply a proof or state explicitly the additional property of S on which the verification relies.
  6. [§4, Lemma 4.2] Lemma 4.2 depends entirely on the cited result [20, Proposition 2.3] that the class of paratopological groups with countable Hausdorff number is closed under arbitrary products and subgroups. This is an acceptable citation, but because the lemma is load-bearing for the paratopological completion theorems, it would improve readability to state the cited proposition explicitly.
  7. [§4, Corollary 4.9] The proof of Corollary 4.9 uses [26, Theorem 3.2] from an arXiv preprint. If the preprint has not yet been published, please provide a published reference or include the statement, since this result is used to conclude R-factorizability of the C-embedded subgroup.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main results are derived from in-paper proofs and independent external theorems, not from their own conclusions.

full rationale

I reviewed the derivation chain and found no step that reduces to its own inputs by construction. Theorem 2.4 is a genuine characterization: it proves the equivalence between simple sm-factorizability and factorization of continuous real-valued functions through strongly submetrizable groups, constructing the required homomorphism and function rather than assuming them. The proof uses only the definition via co-zero sets, countable bases, and a diagonal-product argument. Theorem 3.2 then combines external results ([3, Proposition 5.14] for the bound ib(G) ≤ c, [2, Theorem 5.4.10] for cellularity, [2, Theorem 6.2.2] for Ulam non-measurability of 2^c, and [2, Lemma 8.3.1] for μG = υG) with the in-paper Proposition 3.1, which proves C-embeddedness of a Gδ-dense simply sm-factorizable subgroup by extending continuous functions to the Raĭkov completion. None of the cited items contains the target equality μG = ρωG = υG, so the self-citations here do not load-bear circularly. In Section 4, Lemma 4.2 invokes [20, Proposition 2.3] on preservation of countable Hausdorff number under products and subgroups; that is an external theorem, not a restatement of the paper's conclusion, and it does not presuppose simple sm-factorizability or the completion equalities. The remaining steps in Theorems 4.3, 4.13, and 4.20 follow from Theorem 2.4 and standard realcompactness arguments. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors to force a choice, and the definitions are not self-referential in a way that makes the results true by construction. The paper is therefore self-contained against external benchmarks and its central claims are not circular.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No free parameters or invented entities. The paper introduces the new property 'strongly submetrizable' (Definition 2.1), but it is a defined class of groups, not a postulated entity. The proofs rely on standard and specialized background results, several from the authors' own prior papers; those citations are for established theorems independent of the target claims.

assumptions (9)
  • standard math Every regular paratopological group is completely regular (hence Tychonoff)
    Used in Theorem 4.3 and Lemma 4.4 to ensure realcompactification and the separating family of continuous real-valued functions exist.
  • domain assumption The class of paratopological groups with countable Hausdorff number Hs(G) <= omega is closed under arbitrary products and subgroups
    Cited as [20, Proposition 2.3]; underpins Lemma 4.2 and therefore the diagonal embedding in Theorem 4.3.
  • domain assumption In a Hausdorff paratopological group with Hs(G) <= omega, the G-delta-closure of any subgroup is again a subgroup
    Quoted from [18, Lemmas 3 and 4]; this is what makes the G-delta-closure H in Theorem 4.3 a paratopological group.
  • standard math If Y is a Moscow space and is G-delta-dense in a homogeneous space X, then Y is C-embedded in X
    Used in Proposition 3.1 to extend the factoring function g from F to its G-delta-closure in the Raikov completion.
  • standard math The cardinal number 2^c is Ulam non-measurable, so a space all of whose discrete families of open sets have cardinality at most 2^c satisfies mu X = upsilon X
    Used in Theorems 3.2 and Lemma 4.4 via [2, Theorem 6.2.2] and [2, Lemma 8.3.1].
  • standard math For every topological group H, cellularity satisfies c(H) <= 2^{ib(H)}
    Used in Theorem 3.2 and Lemma 4.4 to bound cardinalities of discrete families of open sets.
  • standard math Every omega-narrow Hausdorff topological group of countable pseudocharacter admits a continuous one-to-one homomorphism onto a separable metrizable group
    Lemma 2.3 relies on this result from [2, Corollary 3.4.25]; it is needed in Theorems 2.4 and 2.15.
  • domain assumption Every regular totally omega-narrow paratopological group is omega-balanced and satisfies the Hausdorff number bound used in Lemma 2.8
    Used in Corollary 2.9 and Theorem 4.13 via [15, Proposition 3.8] and [14, Theorem 2].
  • domain assumption A continuous function from a paratopological group to a regular space remains continuous on its semiregularization
    Used in Theorem 2.4 and Proposition 2.7 via [24, Lemma 3.5].

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Cite this review

Pith. "Pith review of Simply $sm$-factorizable (para)topological groups and their completions." pith.science (2026). https://pith.science/paper/2OQD262D

@misc{pith2026190808627,
  author       = {Pith},
  title        = {Pith review of: Simply $sm$-factorizable (para)topological groups and their completions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OQD262D}},
  note         = {Machine review of arXiv:1908.08627}
}
abstract

Let us call a (para)topological group \emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo\-logical groups by means of continuous real-valued functions. We show that a (para)topo\-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $f\colon G\to \mathbb{R}$, one can find a continuous homomorphism $\varphi$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $g\colon H\to \mathbb{R}$ such that $f=g\circ\varphi$. This characterization is applied for the study of completions of simply $sm$-factorizable topological groups. We prove that the equalities $\mu{G}=\varrho_\omega{G}=\upsilon{G}$ hold for each Hausdorff simply $sm$-factorizable topological group $G$. This result gives a positive answer to a question posed by Arhangel'skii and Tkachenko in 2018. Also, we consider realcompactifications of simply $sm$-factorizable paratopological groups. It is proved, among other results, that the realcompactification, $\upsilon{G}$, and the Dieudonn\'e completion, $\mu{G}$, of a regular simply $sm$-factorizable paratopological group $G$ coincide and that $\upsilon{G}$ admits the natural structure of paratopological group containing $G$ as a dense subgroup and, furthermore, $\upsilon{G}$ is also simply $sm$-factorizable. Some results in [\emph{Completions of paratopological groups, Monatsh. Math. \textbf{183} (2017), 699--721}] are improved or generalized.

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