{"id":"6000f19a-5ec1-408b-97a6-dde0efcbe27a","arxiv_id":"1908.08627","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A (para)topological group is simply sm-factorizable exactly when every continuous real-valued function on it factors through a strongly submetrizable group, and for Hausdorff topological groups in this class the Raikov, Dieudonné, and Hewitt-Nachbin completions agree.","lead":"The paper characterizes a broad class of topological groups, called simply sm-factorizable groups, by the way their continuous real-valued functions factor through nicer groups. It then shows that for these groups three standard completion procedures from topology, the Raikov, Dieudonné, and Hewitt-Nachbin completions, always coincide.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 3.2 is internally coherent and the central argument holds up under scrutiny.","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. The reader identified the paratopological completion results as the weakest assumption, specifically the reliance on [20, Proposition 2.3] for preservation of countable Hausdorff number under products. I agree that this is the least self-contained part of the paper, since the cited result is not proved in the text and is by one of the current authors. However, the preservation property is elementary to verify for products and subgroups, and it is not needed for the central Theorem 3.2, which concerns topological groups rather than paratopological groups. The main theorem's proof is internally consistent: the sm-factorization Theorem 2.4 is proved directly, Proposition 3.1 uses standard facts about Moscow spaces and Gδ-density in a way that checks out, and the cellularity bound c(G) ≤ 2^c combined with the Ulam non-measurability of 2^c correctly yields µG = υG by the cited lemma. I could not identify a load-bearing mathematical flaw, either in the sequence of implications or in the invoked background results. The concrete test I propose would re-examine the single most cited step in the central proof, namely the index-of-narrowness bound, to give additional confidence. Overall, the reader's verdict should remain unchanged.","tokens_in":21837,"tokens_out":42244,"duration_ms":414474,"concrete_test":"As a verification step, independently re-derive the proof of [3, Proposition 5.14] from its cited source and confirm that for every neighborhood U of the identity in a simply sm-factorizable Hausdorff topological group G, the saturated cozero-neighborhood argument yields a subset A of cardinality at most c with A·U = G, so that ib(G) ≤ c. Then confirm that Theorem 5.4.10 of [2] applies to Hausdorff topological groups to give c(G) ≤ 2^{ib(G)}. This directly validates the hinge of Theorem 3.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim, Theorem 3.2, is supported by a coherent chain: Theorem 2.4 provides the factorization through strongly submetrizable groups; Proposition 3.1 supplies C-embedding of G into ρωG using the Moscow-space and Gδ-density arguments; and the bound ib(G) ≤ c together with Ulam non-measurability of 2^c yields µG = υG via the cited lemma. The paragraphs I examined for hidden assumptions, including the extension of π to ρG and the inclusion π(G) ⊆ ρωF, are justified by standard results on dense subgroups and Gδ-density in Tychonoff spaces. The paratopological results in Section 4 are less self-contained because they depend on cited results such as [20, Proposition 2.3] and the unpublished preprint [26], but these are not load-bearing for the main topological-group theorem. I see no concrete mathematical error or missing step that would invalidate the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21907,"tokens_out":57500,"duration_ms":576071,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§2, proof of Theorem 2.4"},{"comment":"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.","section":"§3, Theorem 3.2 and surrounding text"},{"comment":"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.","section":"§4, proof of Theorem 4.3"},{"comment":"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].","section":"§4, proof of Theorem 4.13"},{"comment":"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.","section":"§4, Example 4.17"},{"comment":"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.","section":"§4, Lemma 4.2"},{"comment":"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.","section":"§4, Corollary 4.9"}],"recommendation":"minor_revision","confidential_remarks":"The main theorem is sound and the paper fits the journal's scope. The reliance on [20, Proposition 2.3] and on the unpublished preprint [26] is concentrated in the paratopological part; it is not circular and does not affect the central topological-group theorem, but the editor may wish to ask the authors to make the paratopological part more self-contained. The cross-referencing error between Problems 1.2 and 1.3 should be fixed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real. Theorem 3.2—that every Hausdorff simply sm-factorizable topological group satisfies μG = ρωG = υG and is therefore completion friendly—answers the 2018 Arhangel'skii–Tkachenko question, and the proof holds together. The key step is Proposition 3.1: the Gδ-density argument plus C-embedding via strongly submetrizable images is clean and convincing. I checked the chain from Theorem 2.4 through the ib(G) ≤ c bound and the Ulam non-measurability of 2^c; there is no missing link there.\n\nThe characterization in Theorem 2.4 is genuinely new and useful: simply sm-factorizable is equivalent to factoring every real-valued function through a strongly submetrizable (para)topological group. That is the right way to think about these groups. Theorem 2.15 for ω-narrow groups is a nice addition, and the examples show the authors know where the boundaries are.\n\nThe soft spots are in Section 4. The paratopological results are heavier and less self-contained: Theorem 4.3 depends on Lemma 4.2, which leans on the cited [20, Proposition 2.3] that the class of paratopological groups with countable Hausdorff number is closed under products and subgroups. That is a real external load, and the proof would be more convincing if the key preservation fact were reproduced or at least stated with more context. Also Corollary 4.9 cites the unpublished preprint [26]; this is not fatal, but it is a dependency the reader cannot verify. There are several typos ('hods' for 'holds', 'a simply sm-factorizable' in the abstract) and a few places where separation assumptions could be stated more explicitly. The heavy self-citation is noticeable but not circular—the cited earlier papers do not contain these theorems.\n\nI disagree with any suggestion that the self-citation is a red flag. The paper extends the authors' own framework, and the citations are to the right places. The central argument is not hiding anything.\n\nThis is a specialist paper. Niche, yes, but it answers an explicit open problem and gives a clean characterization. A serious referee should engage with it. I would accept it as a referee assignment, though I would ask the authors to tighten the paratopological section and remove the dependence on the unpublished work if possible.","headline":"Solid paper—the main theorem answers a real open question and the core proof is coherent; worth refereeing.","tokens_in":22450,"tokens_out":1084,"would_cite":true,"duration_ms":11749,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A05","22A30","54H11","54A25","54C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["simply sm-factorizable","paratopological group","topological group","strongly submetrizable","realcompactification","Dieudonné completion","R-factorizable group","Raikov completion"],"falsifier":"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.","tokens_in":21580,"feed_emoji":"🔗","tokens_out":13039,"duration_ms":108192,"temperature":0.7,"pith_summary":"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.","feed_headline":"All three completions coincide for simply sm-factorizable groups","feed_subtitle":"Hausdorff groups in this class have all three major completions coincide and are completion friendly.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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*."],"forward_implications":["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)."],"supporting_citations":[{"why":"Introduced the sm-factorizable hierarchy, proved R-factorizable equals sm-factorizable, and posed the completion problems answered here.","marker":"[3]"},{"why":"Supplies the Raikov completion, Moscow-space C-embedding, cellularity and Ulam-measurability facts used to force μG = υG in Theorem 3.2.","marker":"[2]"},{"why":"Gives the preservation of countable Hausdorff number under products and subgroups used to make the Gδ-closure of the diagonal a subgroup in Lemma 4.2.","marker":"[20]"},{"why":"Semiregularization of a paratopological group is a T3 paratopological group, used to make strongly submetrizable targets regular in Theorem 2.4.","marker":"[12]"},{"why":"Provides open-homomorphism factorization to countable-pseudocharacter groups and strong submetrizability criteria used for weakly Lindelöf paratopological groups.","marker":"[10]"},{"why":"Earlier completion results for paratopological groups whose theorems are improved or generalized; its lemmas give the Gδ-closure subgroup lemma used here.","marker":"[18]"},{"why":"Shows totally ω-narrow paratopological groups are ω-balanced, needed in Corollary 2.9 and Theorem 4.13.","marker":"[15]"},{"why":"Supplies the realcompactness and Gδ-set facts showing the Gδ-closure of the diagonal is realcompact in Theorem 4.3.","marker":"[6]"}],"fun_headline_variants":["Triple completion equality for simply sm-factorizable groups","Answering 2018: all completions coincide for sm-factorizable","Characterization gives triple completion equality","Three completions become one for sm-factorizable groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Triple completion equality for simply sm-factorizable groups","Answering 2018: all completions coincide for sm-factorizable","Characterization gives triple completion equality","Three completions become one for sm-factorizable groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000623,"raw_usage":{"total_tokens":2988,"prompt_tokens":1153,"completion_tokens":1835,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":769,"completion_tokens_details":{"reasoning_tokens":1772}},"tokens_in":769,"tokens_out":1835,"duration_ms":14766,"temperature":1.0,"reasoning_tokens":1772,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:35:43.284505+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Arhangel’skii, M","cited_arxiv_id":null,"evidence_quote":"Introduced the sm-factorizable hierarchy, proved R-factorizable equals sm-factorizable, and posed the completion problems answered here."},{"cited_title":"Arhangel’skii, M","cited_arxiv_id":null,"evidence_quote":"Supplies the Raikov completion, Moscow-space C-embedding, cellularity and Ulam-measurability facts used to force μG = υG in Theorem 3.2."},{"cited_title":"Tkachenko, Embedding paratopological groups into t opological products, Topol","cited_arxiv_id":null,"evidence_quote":"Gives the preservation of countable Hausdorff number under products and subgroups used to make the Gδ-closure of the diagonal a subgroup in Lemma 4.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Semiregularization of a paratopological group is a T3 paratopological group, used to make strongly submetrizable targets regular in Theorem 2.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides open-homomorphism factorization to countable-pseudocharacter groups and strong submetrizability criteria used for weakly Lindelöf paratopological groups."},{"cited_title":"Sanchis, M","cited_arxiv_id":null,"evidence_quote":"Earlier completion results for paratopological groups whose theorems are improved or generalized; its lemmas give the Gδ-closure subgroup lemma used here."},{"cited_title":"Sanchis, M","cited_arxiv_id":null,"evidence_quote":"Shows totally ω-narrow paratopological groups are ω-balanced, needed in Corollary 2.9 and Theorem 4.13."},{"cited_title":"Engelking, General Topology, Heldermann, Berlin (1989)","cited_arxiv_id":null,"evidence_quote":"Supplies the realcompactness and Gδ-set facts showing the Gδ-closure of the diagonal is realcompact in Theorem 4.3."}],"review_version":1}