Transversal, $T_{1}$-independent, and $T_{1}$-complementary paratopological group topologies

Fucai Lin

REVIEW 4 major objections 7 minor 17 references

Transversal, $T_{1}$-independent, and $T_{1}$-complementary paratopological group topologies

T0 review · 4 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any infinite Hausdorff SD-paratopological group—and in particular the Sorgenfrey line—no Hausdorff paratopological group topology is T1-complementary.

desk verdict The Sorgenfrey-line negative answer to Problem 10 is real and worth publishing; the paper also contributes reusable devices (SD-paratopological groups, PT-sequences) for the paratopological group topology lattice, but some extension proofs are sketches and the text needs a cleanup pass. read the letter →

arxiv 1908.06236 v1 pith:JVKFWUHX submitted 2019-08-17 math.GR math.GN

Fucai Lin This is my paper · ORCID

classification math.GRmath.GN MSC 22A0554H1154A2554A3554G20

keywords paratopologicalgroupstransversaltopologiesT1-independentT1-complementarySorgenfreylinePT-sequencePT-filtersubmaximalgrouptopology

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

This paper asks when a group can carry two paratopological group topologies that complement each other in the lattice of topologies: two topologies are T1-complementary when their intersection is the cofinite topology (whose nonempty open sets omit only finitely many points) and their union generates the discrete topology. The main answer is negative for a broad class: if $(G,\tau)$ is an infinite Hausdorff SD-paratopological group, meaning $\tau$ admits a coarser saturated topology $\gamma$ with the same dense subsets, then no Hausdorff paratopological group topology $\sigma$ can be T1-complementary to $\tau$. Because the Sorgenfrey line, the real line with the half-open interval topology, is saturated, this settles an open problem in the negative: the Sorgenfrey line has no T1-complementary Hausdorff paratopological group topology. The paper also builds structural tools, submaximal paratopological group topologies and PT-sequences/PT-filters, that characterize when a topology is transversal and when a sequence-determined paratopological group admits a transversal partner. If correct, the results delimit where complementary paratopological group topologies can exist: complementarity is rare, while transversality, for sequence-determined groups, is common.

What carries the argument

The proofs rest on three mechanisms. (1) The SD-paratopological condition: $(G,\tau)$ is SD if some coarser saturated paratopological topology $\gamma$ has exactly the same dense subsets as $\tau$; saturation means every neighborhood $U$ of the identity has $U^{-1}$ with nonempty interior, and this is the property that converts a hypothetical T1-complement into a dense-set contradiction. (2) The submaximal paratopological group topology $PM_G$, defined as the infimum of all maximal paratopological group topologies on $G$. Theorem 4.2 shows $PM_G$ is a topological group, and Theorem 4.6 uses it as a measuring stick: a non-discrete paratopological group topology $\tau$ admits a transversal paratopological group topology exactly when $\tau$ is not below $PM_G$. (3) PT-filters and PT-sequences: a filter or sequence is PT if some paratopological group topology makes it converge to the identity, and $P(G\mid\phi)$ is the strongest such topology. Theorem 5.6 gives an explicit neighborhood base at 0, the family $\Sigma(\phi^{*})$ of finite-fold sums of tails, and Theorem 5.9 characterizes PT-sequences by asking that each nonzero element be excluded from the finite-sum sets $A(k,m)$. Theorem 5.14 shows $P(G\mid\{a_n\})$ is sequential, and Theorem 5.15 shows that for a nontrivial T-sequence it contains a closed copy of the Arens space $S_2$, hence is not Fréchet–Urysohn.

What would settle it

Exhibit a Hausdorff paratopological group topology $\sigma$ on $\mathbb{R}$ such that, with $\tau$ the Sorgenfrey topology, $\tau\cap\sigma$ is the cofinite topology and $\tau\vee\sigma$ is the discrete topology; such a pair would refute Corollary 3.4 and, with it, Theorem 3.3.

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Extended reading notes

Core claim

The core discovery is that T1-complementarity is impossible for a wide class of paratopological groups, and that transversality can be characterized by a single canonical topology. Theorem 3.3 proves that if $(G,\tau)$ is an infinite Hausdorff SD-paratopological group, then no Hausdorff paratopological group topology $\sigma$ is T1-complementary to $\tau$; the proof shows every $\sigma$-neighborhood of the identity is dense in the coarser saturated topology $\gamma$, so it meets every $\tau$-neighborhood infinitely often, while transversality would require two neighborhoods whose intersection is the singleton identity. Corollary 3.4 applies this to the Sorgenfrey line and answers an open problem negatively. Theorem 3.6 adds a countable-group dichotomy: for a countable infinite group, the absence of T1-independent Hausdorff paratopological group topologies is equivalent to the absence of a transversal T1-independent pair. Theorem 4.6 gives the transversality criterion $(G,\tau)\in PTrans$ iff $\tau$ is not below the submaximal paratopological group topology $PM_G$, and Theorem 4.11 shows transversality lifts from a central subgroup to the ambient non-discrete group. Sections 5 and 6 introduce PT-filters and PT-sequences, characterize when a paratopological group is determined by one, prove that every Hausdorff PT-sequence-determined topology is transversal, show the strongest T-sequence-determined topology admits no T1-complementary Hausdorff paratopological group topology, and prove that a countable group determined by a PT-sequence, when its associated strongest topology is Hausdorff, admits a transversal PT-sequence-determined topology.

Load-bearing premise

Every Hausdorff paratopological group topology on a countable group can be thinned to a coarser Hausdorff paratopological group topology with only countably many basic open sets; if that thinning is impossible for some countable paratopological group, the equivalence in Theorem 3.6 collapses.

Editorial extensions

If this is right

The Sorgenfrey line, and more generally every infinite Hausdorff saturated paratopological group, cannot be part of a T1-complementary pair of Hausdorff paratopological group topologies.
A non-discrete paratopological group topology $\tau$ is transversal exactly when $\tau$ is not contained in the submaximal paratopological group topology $PM_G$, so transversality becomes a comparison with a single canonical topology.
If a non-discrete paratopological group contains a central subgroup that is either discrete non-minimal or admits a transversal paratopological group topology, then the ambient group admits a transversal paratopological group topology.
Every Hausdorff paratopological group topology on an infinite group determined by a PT-sequence is transversal, while the strongest topology determined by a T-sequence admits no T1-complementary Hausdorff paratopological partner.
For countable infinite groups, the absence of T1-independent Hausdorff paratopological group topologies is equivalent to the absence of a transversal T1-independent pair, and a countable group has a non-discrete T1-paratopological group topology exactly when it has a nontrivial PT-sequence.

Reading between the lines

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

A natural next test, not addressed in the paper, is whether saturation alone—without the extra dense-set condition in the definition of SD—already blocks T1-complementarity for every infinite Hausdorff paratopological group; the paper proves the obstruction only for SD groups, though the Sorgenfrey line is saturated.
Because Theorem 5.9 reduces PT-sequencehood to a checkable finite-sum condition, one can search for explicit transversal pairs $P(G\mid\{a_n\})$ and $P(G\mid\{b_n\})$ in concrete countable groups such as $\mathbb{Z}$, using Fibonacci-type or other fast-growing sequences.
If the countable-group equivalence of Theorem 3.6 extended to uncountable groups, then T1-complementarity would reduce entirely to T1-independence at every cardinality; the countable-weight thinning assumption makes that extension nontrivial and testable.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main event is Theorem 3.3 and its Corollary 3.4: the Sorgenfrey line does not admit a T1-complementary Hausdorff paratopological group topology. That answers Problem 10 of the 2017 survey, and I think the proof is correct. The tricky step, where T1-independence forces G\UO finite because UO is open in both topologies, works exactly as the stress-test note says. I checked the product-open logic and it holds. So the central claim is solid.

The paper also gives a nice criterion for transversality in terms of the submaximal paratopological group topology PM_G, proves that PM_G is always a topological group, and introduces PT-sequences and PT-filters as tools. Those are genuinely reusable, and they transplant the Protasov–Zelenyuk machinery into paratopological groups in a plausible way. Theorem 5.9, the characterization of PT-sequences, is a reasonable analogue of the known T-sequence criterion. Credit where due: these are useful working tools, not just a one-off counterexample.

Where it gets softer is the extension material. The proof of Theorem 4.9 is more of a sketch than a proof. The filter base A is asserted to determine a finer non-discrete topology, but it is not shown to be a neighborhood base, and the maximality argument is compressed. Same for Theorem 5.20: the jump from “σ is ω-narrow” to “σ is saturated” relies on the Alas–Sanchis results and is stated too fast. These may well be repairable, but they need real checking.

Two more concerns, both moderate. Theorem 3.6 leans on Shakhmatov’s theorem that every Hausdorff paratopological group topology on a countable group has a coarser Hausdorff paratopological group topology with countable weight. I believe that is in the literature, but since the proof is not included, the equivalence between T1-independence and transversal T1-independence on countable groups stands on that external result. The other issue is the typographical state: “adimit,” “B/suppress laszczyk,” “τ /notlessorslnteqlPM G” and similar corruption appear throughout. It is readable, but it badly needs a proofreading pass.

Bottom line: the central result is correct and the PT-sequence framework is a genuine addition to the paratopological toolbox. The paper should go to a serious referee, with the expectation that the sketched proofs in Sections 4 and 5 get tightened before publication. Specialists in paratopological groups and the lattice of topologies will want to read this. It is not going to reorganize any major branch, but it resolves a posed open problem and gives useful criteria.

Referee Report

4 major / 7 minor

Summary. The paper studies transversality, T1-independence, and T1-complementarity for paratopological group topologies. Its main headline result is Theorem 3.3 and Corollary 3.4: no infinite Hausdorff SD-paratopological group, and in particular the Sorgenfrey line, admits a T1-complementary Hausdorff paratopological group topology, giving a negative answer to Problem 3.2 (Problem 10 of [5]). The paper also introduces submaximal paratopological group topologies (Theorems 4.2 and 4.6), an extension theorem for central subgroups (Theorem 4.11), and develops PT-sequences and PT-filters, proving a characterization of Abelian paratopological groups determined by PT-sequences (Theorem 5.9), a sequentiality/non-Fréchet-Urysohn result (Theorems 5.14 and 5.15), the nonexistence of T1-complementary topologies for the strongest topology determined by a T-sequence (Theorem 5.20), and several results on countable groups and PT-filters in Section 6.

Significance. If the main theorem is correct, it resolves Problem 10 of [5] in the negative, a result that is likely to be of interest to researchers working on paratopological groups and lattice-theoretic complementarity. I have checked the proof of Theorem 3.3 in detail and found it sound: the claim that UO is open in both σ and τ is valid because left and right translations are homeomorphisms in a paratopological group, the use of T1-independence to conclude that G\UO is finite is legitimate, and the final density/transversality contradiction is correct. The paper also offers useful new tools, especially the submaximal paratopological group topology and PT-sequences. However, several other advertised results, specifically Theorems 3.5, 3.6, 4.9, and 5.20, contain proof gaps or misquotations that must be addressed before the paper can be accepted.

major comments (4)

[Section 4 (Theorem 4.9)] The proof of Theorem 4.9 is incomplete. In the case where N = ker f is neither τ-open nor τ-discrete, the assertion that "it is obvious that one can get a finer non-discrete paratopological group topology on G by taking N open" is not justified: one must check that the topology generated by adding N as a neighborhood of e is a paratopological group topology, and that it is not discrete. In the discrete case, the family A is asserted to be a filter base determining a non-discrete paratopological group topology η with η > τ, but the axioms for a neighborhood base of a paratopological group topology are not verified, and it is not shown that η is strictly finer than τ or that η is non-discrete. Because Theorem 4.9 is used in the proof of Theorem 4.11 and Corollary 4.12, these omissions affect the central-subgroup extension result advertised in the abstract.
[Section 3 (Theorem 3.5)] Theorem 3.5 is stated with "it easily prove" after the proof of Theorem 3.3, but the argument of Theorem 3.3 does not adapt directly. In the complementary (rather than T1-complementary) setting, a nonempty set UO open in both σ and τ must be equal to G because the intersection of the two topologies is indiscrete, so the contradiction G\UO finite used in Theorem 3.3 is not available. The authors should provide a complete proof of Theorem 3.5 or remove the claim.
[Section 3 (Theorem 3.6)] In the proof of Theorem 3.6, the passage from a Hausdorff paratopological group topology τ_i to a coarser Hausdorff paratopological group topology σ_i of countable weight relies on Shakhmatov's result [13] and [2, Problem 5.2.A]. Since [2] is mainly about topological groups, it is not clear that the quoted statement applies verbatim to paratopological groups, where inversion need not be continuous. Please state the exact result and explain why it holds for paratopological groups; if it does not, the equivalence in Theorem 3.6 is unproved, although Theorem 3.3 is unaffected.
[Section 5 (Proposition 5.19 and Theorem 5.20)] The statement of Proposition 5.19 has the topologies swapped relative to its use in Theorem 5.20. As printed, the proposition says that if σ is sequential then the space (G,σ) is countably compact and has no nontrivial convergent sequences; in Theorem 5.20 it is τ = P(G|{a_n}) that is sequential, so the printed proposition does not justify the conclusion that (G,σ) is countably compact. Please correct the statement of Proposition 5.19 or the proof of Theorem 5.20, and verify the quotation from [15, Proposition 2.4].

minor comments (7)

[Throughout] There are typographical errors such as "adimit" (Abstract and Corollary 3.4), "transveral" (Abstract), and "B/suppress laszczyk" (Introduction); these should be corrected.
[Section 3 (Theorem 3.3 proof)] In the final transversal step of the proof of Theorem 3.3, the statement should read "there exist U∈σ(e) and V∈τ(e) such that U∩V={e}", not "there exist U∈σ(e) and V∈σ(e)", and the identity element should be denoted consistently as e rather than 0.
[Section 4 (Theorem 4.6)] In Theorem 4.6 and its proof, the expression "τ /notlessorslnteql PM G" is garbled; it should be "τ ≰ PM_G".
[Section 5 (Theorem 5.14 proof)] In the proof of Theorem 5.14, "0 ∈ F \ F" should read "0 ∈ \overline{F} \ F".
[Section 5 (notational consistency)] The notation P(G|{a_n}) is used with the index set ω in most of Section 5, but Theorem 5.20 uses {a_n}_{n∈N}; please standardize the indexing throughout.
[Section 6 (Theorem 6.12, Definition 6.3)] In Theorem 6.12, "there exits" should be "there exists", and in Definition 6.3 the notation "{Dn}nω" should be "{D_n}_{n∈ω}".
[Abstract] The abstract says "a non-discrete paratopological group topology G contains a central subgroup", but the corresponding theorem is about a group H containing a subgroup G; please rephrase to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the Sorgenfrey-line theorem is derived from definitions and independent prior results.

full rationale

The derivation chain is self-contained. Theorem 3.3 introduces the sufficient condition of being an SD-paratopological group, then proves that if a T1-complementary Hausdorff paratopological group topology existed, a neighborhood U of the identity in the second topology would be dense in the first topology; T1-independence forces every nonempty set open in both topologies to have finite complement, so any pair of identity neighborhoods meets infinitely often, contradicting transversality. The SD condition is not an encoding of the conclusion: it supplies a saturated coarser topology, and the rest of the argument uses only the T1-independence and transversality hypotheses. Corollary 3.4 is a direct application to the Sorgenfrey line, which is saturated. The external theorems invoked (Shakhmatov's theorem in Theorem 3.6, results from [5] and [11] in later sections) are genuine prior results and do not restate the theorem under proof. The single self-citation, [9, Theorem 3.8] in Theorem 3.7, is ancillary to the main Sorgenfrey-line claim and refers to an earlier published proof rather than to an assumption equivalent to the target result. No parameter is fitted and no prediction is merely an input under a new name, so no circularity is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper is a pure mathematics argument in ZFC. It introduces no fitted constants. The load-bearing external inputs are Shakhmatov's second-countable refinement theorem [13], the Protasov-Zelenyuk theory of T-sequences [11], and the Alas-Sanchis saturation theorem [1]; each is cited but not proved. The paper's own new definitions, SD-paratopological groups, PT-sequences, and PM_G, are internal constructs with no independent empirical evidence, so they are listed as invented entities with independent_evidence false.

assumptions (5)

standard math Zorn's lemma
Used in Section 4 to guarantee that every non-discrete paratopological group topology extends to a maximal one, which defines PM_G and underlies Theorem 4.6.
domain assumption Shakhmatov's theorem on condensations with countable weight
Invoked in Theorem 3.6 to replace each Hausdorff paratopological group topology on a countable group by a coarser Hausdorff paratopological group topology of countable weight; this is an external result whose proof is not given.
domain assumption Protasov and Zelenyuk theory of T-sequences and topologies T(G|phi)
Sections 5 and 6 rely repeatedly on [11], including the existence of nontrivial T-sequences, Lemma 2.1.6, Theorem 2.2.7, and the construction of a transversal T-sequence topology used in Theorem 6.14.
domain assumption Alas and Sanchis theorem on countably compact paratopological groups
Theorem 5.20 uses [1, Theorems 2.1 and 2.5] to infer that a countably compact, omega-narrow paratopological group is saturated; this is essential to the non-existence of a T1-complementary topology for P(G|{a_n}).
domain assumption Dikranjan, Tkachenko, and Yaschenko foundation for transversal group topologies
Definitions 1.1 and 1.2 and prior results on transversal group topologies from [6] and [15,17] are assumed as the starting framework; the paper does not re-derive them.

invented entities (3)

SD-paratopological group
purpose: Encodes a sufficient condition, a saturated topology below the given topology preserving dense sets, to prove Theorem 3.3.
New class defined in the paper; its only support is the proof of Theorem 3.3, so it has no external falsifiable handle.
PT-sequence and PT-filter
purpose: Sequences or filters that converge in some T1 paratopological group topology; used to construct the strongest topology P(G|phi) and characterize transversality.
Internal mathematical definitions adapted from T-sequences [11]; they produce theorems inside the paper but no observable predictions outside it.
Submaximal paratopological group topology PM_G
purpose: Infimum of all maximal paratopological group topologies; gives the transversality criterion that a topology belongs to PT rans exactly when it is not below PM_G.
Defined in Section 4; supported only by the proofs that use it, with no independent evidence outside the paper.

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Pith. "Pith review of Transversal, $T_{1}$-independent, and $T_{1}$-complementary paratopological group topologies." pith.science (2026). https://pith.science/paper/JVKFWUHX

@misc{pith2026190806236,
  author       = {Pith},
  title        = {Pith review of: Transversal, $T_1$-independent, and $T_1$-complementary paratopological group topologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JVKFWUHX}},
  note         = {Machine review of arXiv:1908.06236}
}

abstract

We discuss the class of paratopological groups which admits a transversal, $T_{1}$-independent and $T_{1}$-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a $T_{1}$-complementary Hausdorff paratopological group topology, which gives a negative answer to \cite[Problem 10]{AT2017}. We give a very useful criterion for transversality in term of submaximal paratopological group topology, and prove that if a non-discrete paratopological group topology $G$ contains a central subgroup which admits a transversal paratopological group topology, then so does $G$. We introduce the concept of $PT$-sequence and give a characterization of an Abelian paratopological group being determined by a $PT$-sequence. As the applications, we prove that the Abelian paratopological group, which is endowed with the strongest paratopological group topology being determined by a $T$-sequence, does not admit a $T_{1}$-complementary Hausdorff paratopological group topology on $G$. Finally, we study the class of countable paratopological groups which is determined by a $PT$-filter, and obtain a sufficient condition for a countable paratopological group $G$ being determined by a $PT$-sequence which admits a transversal paratopological group topology on $G$ being determined by a $PT$-sequence.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

B/suppress laszczyk, M

A. B/suppress laszczyk, M. Tkachenko,Transversal, T1-independent, and T1-complementary topologies , Topology Appl., 230(2017): 308–337

work page 2017

[2] [13]

Shakhmatov, Condensations of universal topological algebras preservi ng continuity of operations and decreasing weights, Moscow Univ

D.B. Shakhmatov, Condensations of universal topological algebras preservi ng continuity of operations and decreasing weights, Moscow Univ. Bull., 39(2)(1984): 57–60. Russian original in: Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2(1984): 42–45

work page 1984

[3] [2]

Arhangel’skiˇ ı, M

A. Arhangel’skiˇ ı, M. Tkachenko, Topological groups and related structures , Atlantis Press, Paris; World Scientiﬁc Publishing Co. Pte. Ltd., Hackensack, NJ, 2008

work page 2008

[4] [1]

O.T. Alas, M. Sanchis, Countably compact paratopological groups, Semigroup Forum, 74(2007): 423–438

work page 2007

[5] [3]

Bagley, On the characterization of the lattice of topo logies, J

R. Bagley, On the characterization of the lattice of topo logies, J. London Math. Soc. 30 (1955) 247C249

work page 1955

[6] [4]

Birkhoﬀ, On the combination of topologies , Fundam

G. Birkhoﬀ, On the combination of topologies , Fundam. Math., 26(1936): 156–166

work page 1936

[7] [6]

Dikranjan, M

D. Dikranjan, M. Tkachenko, I. Yaschenko, On transversal group topologies , Topology Appl., 153(2005): 786–817

work page 2005

[8] [7]

Dikranjan, M

D. Dikranjan, M. Tkachenko, I. Yaschenko, Transversal group topologies on non-Abelian group , Topology Appl., 153(2006): 3338–3354

work page 2006