Normality in the square of the Sorgenfrey Line

Hongwei Wen; Paul Szeptycki

arxiv: 2511.12327 · v2 · submitted 2025-11-15 · 🧮 math.GN

Normality in the square of the Sorgenfrey Line

Paul Szeptycki , Hongwei Wen This is my paper

Pith reviewed 2026-05-17 21:59 UTC · model grok-4.3

classification 🧮 math.GN

keywords Sorgenfrey topologynormalitylambda-setsQ-setscontinuum hypothesissets of realspseudo-normality

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The pith

Under CH there exists a non-λ-set of reals whose Sorgenfrey square is normal.

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

The paper studies normality of squares in the Sorgenfrey lower-limit topology on subsets X of the reals. Przymusiński showed that every Q-set produces a normal square. The authors prove that every λ-set produces a pseudo-normal square. They then show that the converse directions fail: assuming the continuum hypothesis, there is a set concentrated on a countable dense subset that is not a λ-set yet still has a normal Sorgenfrey square. This separates the topological property of the square from the classical set-theoretic properties of X.

Core claim

Assuming the continuum hypothesis, there exists a set X of reals concentrated on a countable dense subset (hence not a λ-set) such that the square (X[≤])² is normal in the Sorgenfrey lower-limit topology.

What carries the argument

The Sorgenfrey lower-limit topology X[≤] on a set of reals and the question of when its square is normal or pseudo-normal.

If this is right

Normality of (X[≤])² does not imply that X is a λ-set.
The square can be normal for sets that are concentrated on countable dense sets.
Pseudo-normality holds for all λ-sets while full normality can hold for some strictly larger class under CH.

Where Pith is reading between the lines

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

Concentration on a countable dense set may be enough to control the square's separation properties once CH supplies the right set.
The gap between λ-sets and the normality condition invites similar questions for other separation axioms in the Sorgenfrey topology.

Load-bearing premise

The continuum hypothesis is assumed to produce a set concentrated on a countable dense subset that fails to be a λ-set.

What would settle it

An explicit construction of the set X under CH together with a direct check that (X[≤])² satisfies the definition of normality would confirm the claim; showing no such X exists even under CH would refute it.

read the original abstract

We consider sets of reals $X$ endowed with the Sorgenfrey lower limit topology denoted $X[\leq]$. Przymusi\'nski proved that if $X$ is a $Q$-set then $(X[\leq])^2$ is normal. While the converse is not in general true we consider examples of sets of the reals for which $(X[\leq])^2$ is normal or just pseudo-normal. For example, if $X$ is a $\lambda$ set, then $(X[\leq])^2$ is pseudo-normal but assuming CH there is an $X$ concentrated on a countable dense subset (so not a $\lambda$-set) but still $(X[\leq])^2$ is normal.

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.

Desk Editor's Note private letter to a colleague

Under CH this paper gives a set X concentrated on a countable dense subset that fails to be a λ-set yet has normal Sorgenfrey square.

read the letter

The central point is that the authors produce, assuming CH, a set X of reals that is concentrated on a countable dense subset and therefore not a λ-set, but whose Sorgenfrey square is nevertheless normal. This separates the normality property from the λ-set condition in a way that goes beyond the known implication from Q-sets to normal squares and the pseudo-normality that λ-sets deliver. The construction appears to rely on standard CH techniques for building concentrated sets that avoid the λ property while preserving enough control for the square to be normal. That distinction is the useful piece of work here. The paper does a clean job of stating the background from Przymusiński and then positioning the new example against both the Q-set and λ-set cases. The abstract makes the logical separation explicit without overclaiming. The main limitation is the dependence on CH. The example is consistent rather than provable in ZFC, which is typical for these kinds of separating results but means the result does not give an unconditional counterexample. Without the full proofs it is impossible to check whether the normality verification handles all the relevant open sets correctly or whether any auxiliary claims about the topology are fully justified. Still, the overall shape of the argument looks proportionate to what is claimed. This is a paper for set-theoretic topologists who already know the Sorgenfrey line and the basic facts about Q-sets and λ-sets. A reader working on normality questions in non-metrizable spaces or on consistency results for topological properties will find the separating example directly relevant. It is narrow enough that it will not interest a broad audience, but within its niche the contribution is clear. The work is coherent on its own terms and engages the existing literature without obvious internal contradictions. I would send it to a serious referee rather than desk-reject it.

Referee Report

0 major / 2 minor

Summary. The paper studies normality and pseudo-normality of the square of the Sorgenfrey lower-limit topology on subsets X of the reals. It recalls Przymusiński's theorem that every Q-set yields a normal square, notes that the converse fails in general, and supplies two families of examples: every λ-set produces a pseudo-normal square, while under CH there exists a set X concentrated on a countable dense subset (hence not a λ-set) whose square is nevertheless normal.

Significance. The construction separates normality of (X[≤])² from the λ-set and Q-set properties, sharpening the boundary between these notions in the Sorgenfrey setting. The explicit CH construction and the contrast with the λ-set case (which reaches only pseudo-normality) constitute the main technical contribution.

minor comments (2)

The notation X[≤] is introduced without an explicit reminder of the definition of the Sorgenfrey topology; a one-sentence recall in §1 would aid readers unfamiliar with the lower-limit topology.
The abstract states the CH example but does not indicate the length or location of the construction; a forward reference to the relevant section would improve navigation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive recommendation to accept. The referee's summary accurately captures the main results, including the separation between normality of the Sorgenfrey square and the λ-set property under CH.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs, under CH, an explicit set X of reals that is concentrated on a countable dense subset yet fails to be a λ-set, while verifying that (X[≤])² remains normal. This is a direct existence proof via set-theoretic methods rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The cited Przymusiński theorem is an external result establishing the Q-set implication, and the CH hypothesis is invoked only to ensure the existence of the desired counterexample; no step reduces the claimed normality to the input assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central existence claim rests on the continuum hypothesis together with standard definitions of Q-sets, λ-sets, and concentrated sets; no free parameters or new entities are introduced.

axioms (1)

domain assumption Continuum Hypothesis (CH)
Invoked to obtain the existence of a concentrated set X that fails to be a λ-set.

pith-pipeline@v0.9.0 · 5414 in / 1041 out tokens · 34431 ms · 2026-05-17T21:59:30.672429+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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A Lindelöf spaceXsuch thatX2 is not Lindelöf.Fundamenta Math- ematicae, 78(Issue Number):291–296, 1973

T.Przymusinski. A Lindelöf spaceXsuch thatX2 is not Lindelöf.Fundamenta Math- ematicae, 78(Issue Number):291–296, 1973

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[8]

E. K. van Douwen. The integers and topology. InHandbook of Set-theoretic Topology, pages 111–167. North-Holland, Amsterdam, 1984. NORMALITY IN THE SQUARE OF THE SORGENFREY LINE 13 Department of Mathematics and Statistics, York University, 4700 Keele St, Toronto, ON, Canada, M3J 1P3. Email address:szeptyck@yorku.ca Department of Mathematics and Statistics,...

work page 1984

[1] [1]

Baumgartner

J. Baumgartner. Allℵ 1-dense sets of reals can be isomorphic.Fundamenta Mathe- maticae, 79(2):101–106, 1973

work page 1973

[2] [2]

R. H. Bing. Metrization of topological spaces.Canadian Journal of Mathematics, 3:175–186, 1951

work page 1951

[3] [3]

Hausdorff

F. Hausdorff. Probleme 58.Fundamenta Mathematicae, 20:286, 1933

work page 1933

[4] [4]

F. D. Tall. Normality versus collectionwise normality. InHandbook of set-theoretic topology, pages 685–732. North-Holland, Amsterdam, 1984

work page 1984

[5] [5]

Todorčević

S. Todorčević. Remarks on chain conditions in products.Compositio Mathematica, 55(3):295–302, 1985

work page 1985

[6] [6]

Todorcevic.Partition Problems in Topology, volume 84 ofContemporary Mathe- matics

S. Todorcevic.Partition Problems in Topology, volume 84 ofContemporary Mathe- matics. American Mathematical Society, Providence, RI, 1989

work page 1989

[7] [7]

A Lindelöf spaceXsuch thatX2 is not Lindelöf.Fundamenta Math- ematicae, 78(Issue Number):291–296, 1973

T.Przymusinski. A Lindelöf spaceXsuch thatX2 is not Lindelöf.Fundamenta Math- ematicae, 78(Issue Number):291–296, 1973

work page 1973

[8] [8]

E. K. van Douwen. The integers and topology. InHandbook of Set-theoretic Topology, pages 111–167. North-Holland, Amsterdam, 1984. NORMALITY IN THE SQUARE OF THE SORGENFREY LINE 13 Department of Mathematics and Statistics, York University, 4700 Keele St, Toronto, ON, Canada, M3J 1P3. Email address:szeptyck@yorku.ca Department of Mathematics and Statistics,...

work page 1984