On some modifications of the Niemytzki plane

Wojciech Bielas

REVIEW 4 major objections 4 minor 7 references

On some modifications of the Niemytzki plane

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that replacing the Niemytzki plane's tangent discs with triangles whose opening angles stay bounded away from zero produces a finer topology that is not homeomorphic to the original plane.

desk verdict New criterion for Niemytzki-type topologies; the main non-homeomorphism result is sound, but the proof needs a filled-in boundary-invariance argument and a quantifier fix. read the letter →

arxiv 2506.01707 v1 pith:LOOZ4TRD submitted 2025-06-02 math.GN

Wojciech Bielas This is my paper

classification math.GN MSC 54C3054G2054D7026A24

keywords Niemytzkiplanetopologymodificationsbasicfamilyoffunctionsbounded-angletriangleshomeomorphismcriterionmonotoneLebesguedifferentiationtheoremreal

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 claims that the topology of the Niemytzki plane is sensitive to the shapes of the tangent neighborhoods used to define it. It works with 'basic families'—even hump-shaped functions $f_n$ on shrinking intervals that generate a topology on the closed upper half-plane by adding translated tangent sets to the Euclidean topology—and proves a necessary condition for two such topologies to be homeomorphic. The condition forces any homeomorphism to induce a monotone boundary map $g:\mathbb{R}\to\mathbb{R}$ and functions $\gamma,\delta$ with $\liminf\gamma\le 1$ satisfying a quantitative inequality comparing the inverse shapes of the two families. Applying this criterion to tangent parabolas (the Niemytzki plane) and to triangles $t_{\alpha,n}(z)=|z|\tan(\alpha n/(n+1))$ with vertex angles bounded away from zero produces a contradiction, so the triangle topology is finer and not homeomorphic to the Niemytzki plane.

What carries the argument

The central object is a basic family $\{f_n\}$: even continuous functions $f_n:[-a_n,a_n]\to[0,1/n]$ with $f_n(0)=0$, strictly monotone on each side of $0$, whose Euclidean closures are nested, so that the translated sets $U(x,f_n)$ form a base at each boundary point. The load-bearing mechanism is the comparison inequality of Lemma 2. After a homeomorphism is forced to send the boundary line to itself and to act there as a monotone map $g$, the Baire category theorem freezes indices $n,k$ on intervals, and Proposition 1, proved via the Jordan curve theorem, bounds the ordinate of the image of the midpoint of two overlapping neighborhoods. Combining the two bounds yields $t_1(\delta(x))\le t_k\!\big((p_m^{-1}(x)/p_n^{-1}(x))\,\delta(x)\gamma(x)\big)$ with $\liminf_{x\to0^+}\gamma(x)\le1$, where $\gamma$ is a ratio of symmetric difference quotients of $g$. The bound on $\gamma$ is where the Lebesgue monotone differentiation theorem enters: monotone functions are differentiable a.e., and when $g'=0$, Lemma 3 controls the ratio of quotients through continuity.

What would settle it

A direct test is to attempt to construct a homeomorphism between the bounded-angle triangle topology and the Niemytzki plane explicitly near a boundary point. Lemma 2 forces the boundary homeomorphism $g$ to produce functions $\gamma,\delta$ with $\liminf\gamma\le1$ satisfying $t_1(\delta(x))\le t_k\!\big(\sqrt{n/m}\,\delta(x)\gamma(x)\big)$ for all small $x$; for fixed $k,n$, choosing $m$ so large that $\sqrt{n/m}\,\tan\alpha/\tan(\alpha/2)<1$ makes the inequality fail along a sequence, so exhibiting any $g$ whose $\gamma$ has $\liminf\le1$ while satisfying the inclusion conditions would refute the paper. Conversely, computing $\liminf\gamma$ for a proposed boundary map and seeing it exceed $1$ would confirm the obstruction.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that shape matters for the Niemytzki plane: the topology depends not just on the fact that tangent sets are small and flatten toward the boundary, but on whether their opening angles have a positive lower bound. The standard tangent discs, the parabolas of Example 1, and all power-shaped families $p_{s,n}(x)=n|x|^s$ produce spaces homeomorphic to the Niemytzki plane; the triangles $t_{\alpha,n}(z)=|z|\tan(\alpha n/(n+1))$, whose half-angles $\alpha n/(n+1)$ increase to $\alpha$ and whose vertex angles therefore stay in $[\alpha,2\alpha)$, produce a finer topology that is not homeomorphic to it. The proof works by deriving the comparison criterion of Lemma 2 from the a.e. differentiability of the monotone boundary homeomorphism, then showing that for bounded-angle triangles every possible choice of $\gamma,\delta$ violates the criterion.

Load-bearing premise

The load-bearing premise is that for every $m>n$ the sets $(\operatorname{cl}U(0,p_n))\setminus U(0,p_m)$ and their counterparts in the target space are non-compact, so every homeomorphism must map the boundary line onto itself and therefore be represented by a monotone function $g:\mathbb{R}\to\mathbb{R}$; if boundary invariance ever failed, the proof of Lemma 2 would not get started.

Editorial extensions

If this is right

For every $\alpha\in(0,\pi/2)$, the modified Niemytzki plane built from $t_{\alpha,n}(z)=|z|\tan(\alpha n/(n+1))$ is not homeomorphic to the Niemytzki plane, and its topology is strictly finer than the Niemytzki topology.
The necessary criterion of Lemma 2 applies to any two basic families, so the same argument can certify non-homeomorphism for other shapes, as the paper does for $w_n(x)=|x|^{(n+1)/n}$, which is neither the Niemytzki plane nor the bounded-angle triangle space.
Any homeomorphism between two such spaces must induce a monotone boundary map whose symmetric difference quotients obey inequality (1); hence the differentiability theory of monotone functions imposes a quantitative rigidity on topological equivalence in this class.
The chain $\tau_1\subsetneq\tau_2\subsetneq\tau_3$ with $\tau_1\cong\tau_3$ shows there are at least two distinct modified Niemytzki planes between the usual topology and the 'thin triangle' topology, so the family of spaces between the Euclidean and Niemytzki topologies is richer than previously recorded.

Reading between the lines

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

The criterion suggests a possible classification of modified Niemytzki planes by the asymptotic behavior of the inverse sizes $f_n^{-1}(x)$; shapes whose inverse functions are comparable in the ratio sense of Lemma 2 may be homeomorphic, and bounded-angle cones may form a distinct class. The paper does not state such a classification.
The proof's reliance on a.e. differentiability hints that any homeomorphism between two such spaces is almost linear at small scales on a set of positive measure; if true, such homeomorphisms cannot be completely wild. This is an inference, not a claim of the paper.
A natural testable extension is to replace 'bounded away from zero' with a slower decay of the vertex angle, for example $\theta_n\to0$ at a controlled rate; the paper leaves open whether such intermediate families are homeomorphic to the Niemytzki plane or to the bounded-angle triangles.

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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

Jan, quick take on Bielas's arXiv:2506.01707. This is a solid, genuinely new piece of general topology. The central result—that replacing the tangent discs in the Niemytzki plane with triangles having bounded vertex angles gives a non-homeomorphic space—is established through a new quantitative criterion (Lemma 2) that compares two basic families. That criterion is the real contribution. The paper also shows in Example 2 that different parabola shapes all produce homeomorphic copies of the Niemytzki plane, which is a helpful contrast and rules out the suspicion that any shape change alters the topology.

The proof architecture is convincing. Lemma 2 reduces any homeomorphism to an inequality involving derivative quotients, and the applications in Examples 3 and 4 use that inequality in a clean way. The reliance on Jordan curve theorem, Baire category, and Lebesgue monotone differentiation is appropriate, and the criterion is verified against the classical space. No fitted parameters, no circularity, no self-citation issues.

Now the soft spots. Both are real but repairable. First, the boundary-invariance step in Section 3 is load-bearing and presented as a one-line 'Note that.' The proof needs every homeomorphism to preserve the boundary line, and the text only says the crescent sets are non-compact, hence f2(x,0)=0. That implication is not obvious. The missing bridge is local compactness: boundary points have no compact neighborhoods because any basic neighborhood U(x,p_n) has a closed, non-compact crescent inside its closure, while interior points have compact Euclidean neighborhoods. This is standard and almost certainly true, but it must be written out. Second, Example 3 has a quantifier slip: it fixes k before choosing m, while Lemma 2 gives k depending on m. The contradiction still works if you instead choose m large enough and use tan(alpha k/(k+1)) <= tan alpha, but as written the logic is off. The non-compactness assertion itself is also stated without proof.

None of these sink the paper. The main theorem is very likely correct, and the gaps are exactly the kind a referee would demand be filled. This deserves a serious referee, not a desk reject. I'd send it out with a request to expand the boundary-invariance proof and tidy Example 3. For a topologist working with counterexamples, the criterion is a useful addition; I'd bring it to our reading group.

Referee Report

4 major / 4 minor

Summary. The paper studies modifications of the Niemytzki plane obtained by replacing the usual tangent disks with sets of the form U(x,f_n), where {f_n} is a "basic family" of even functions. The main tool is Lemma 2, a necessary condition for two such modified topologies to be homeomorphic, expressed through an inequality involving derivative quotients of a boundary map g. The paper uses this criterion in Example 3 to show that the topology induced by triangles with a fixed bounded angle is not homeomorphic to the usual Niemytzki plane, and in Example 4 to distinguish several other power-type families. The proof strategy is to reduce a hypothetical homeomorphism to a monotone function g on the boundary, apply the Lebesgue differentiation theorem to obtain differentiability at some point, and then compare tangent-neighborhood shapes through the inequality in Lemma 2.

Significance. If the main result is correct, the paper gives a clean quantitative criterion for distinguishing Niemytzki-plane modifications and answers a natural shape-dependence question: changing tangent disks to fixed-angle triangles changes the topology. The paper is elementary and self-contained, relying on standard tools (Jordan curve theorem, Baire category theorem, Lebesgue monotone differentiation theorem), and it produces explicit non-homeomorphic spaces with a transparent proof architecture. The criterion in Lemma 2 could be useful for further systematic study of tangent-neighborhood modifications. However, the current presentation contains load-bearing gaps in the boundary-invariance step and in the quantifier structure of Lemma 2; these must be repaired before the main claims are fully established.

major comments (4)

[Section 3, after Lemma 3] The sentence "Note that sets (cl U(0,p_n))\U(0,p_m) and their t-counterparts are not compact for any m>n, hence f2(x,0)=0 for any x∈R" is a load-bearing step that is not proved. Non-compactness of these crescents is asserted for the modified topology, although in the Euclidean topology the crescents are compact closed subsets of cl U(0,p_n); the proof must explain why the finer topology destroys compactness. Moreover, even if each crescent is non-compact, the conclusion that every homeomorphism sends the boundary line into itself requires a characterization of boundary points, for example via local compactness, showing that boundary points have no compact neighborhoods while interior points do. Since the function g is defined only after this step, and Lemma 2, Example 3 and Example 4 all depend on it, this gap must be closed.
[Lemma 2, Section 3] The statement of Lemma 2 asserts the existence of a single pair of functions γ,δ such that for every m>n there exists k satisfying the displayed inequality. In the proof, however, m is fixed before γ is defined: γ(w)=I(u,p_m^{-1},w)/I(u,p_n^{-1},w), so γ depends on m. Thus the lemma as stated is stronger than what is actually proved. The applications in Examples 3 and 4 only need the weaker form in which δ is fixed and γ is allowed to depend on m, but the statement and proof must be aligned, or Lemma 2 should be rewritten with explicit quantifiers indicating which objects are uniform in m and which are not.
[Example 3, Section 3] The contradiction argument in Example 3 fixes k before choosing m>n, whereas Lemma 2 only provides k after m is chosen. This reverses the quantifier order and, as written, does not contradict the lemma. The argument can be repaired because tan(α k/(k+1)) < tan α for every k, so one may first choose m with √(n/m) tan α / tan(α/2) < 1 and then use the k supplied by the corrected Lemma 2; nevertheless, the present text should be revised to reflect this order.
[Proposition 1(b), Section 2] The proof of Proposition 1(b) is not supplied; the text says it "can be verified by examining the above picture." This bound is used in the derivation of inequalities (3) and (5) in the proof of Lemma 2, so it is load-bearing for the main criterion. The relevant comparison of ordinates should be proved analytically, for instance by using the monotonicity of f_n and the Jordan curve theorem, rather than left to a diagram.

minor comments (4)

[Lemma 2, Section 3] The placement of the limit conditions is ambiguous; "lim_{h→0}δ(h)=0 and lim inf_{h→0}γ(h) ≤1" should be stated as part of the existence claim for δ and γ, before the quantifier "for every m>n," to avoid confusion about what is uniform.
[Example 3, Section 3] The phrase "substituting z = p_m^{-1}(x)/p_n^{-1}(x) δ(x)γ(x), in the counter" should read "in the numerator".
[Section 3, after Lemma 3] The assertion about non-compactness of the crescent sets should explicitly say that compactness is meant in the modified topology, because in the Euclidean topology these sets are compact.
[Lemma 4, Section 3] The proof of Lemma 4 is largely descriptive and relies on arcs and regions "as in the picture." The connectivity claims, especially that points in certain regions can be connected by arcs avoiding the relevant closed sets, should be formalized, since the monotonicity of g is central to the subsequent differentiation argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-homeomorphism result is derived from a proven necessary condition using external standard theorems, with no fitted inputs or self-cited premises.

full rationale

The paper's derivation is self-contained: it defines basic families, proves Proposition 1 using the Jordan curve theorem, proves Lemma 3 by an elementary continuity argument, proves Lemma 4 by a planar arc argument, and then proves Lemma 2 as a necessary condition for a homeomorphism between two such topologies. Example 3 applies Lemma 2 to the triangle family t_alpha,n and the parabola family p_n, derives an inequality, and obtains a contradiction from lim inf <= 1. None of these steps assumes the conclusion that the spaces are not homeomorphic; the contradiction is produced, not presupposed. No parameter is fitted to data, no predicted quantity is defined in terms of the observed outcome, and no load-bearing result is imported from the author's own prior work. The one notable gap is the asserted boundary-line invariance f2(x,0)=0 after Lemma 3, supported only by non-compactness of the crescent sets; this is a potential rigor gap about local compactness, not a circular reduction. Under the hard rules, missing justification for a true topological assertion is a correctness concern, not circularity. The paper uses only standard external theorems: Jordan curve theorem, Baire category theorem, and Lebesgue monotone differentiation theorem. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The proof relies on standard theorems of topology and real analysis, plus one asserted geometric property of the constructed spaces (non-compactness of deleted boundary neighborhoods) that the paper does not prove in detail.

assumptions (4)

standard math Jordan curve theorem
Used in Proposition 1 to identify the bounded and unbounded components of the complement of two translated tangent neighborhoods, and in Lemma 4 for arc components.
standard math Baire Category Theorem
Used in Lemma 2 twice to pass from pointwise indices n_x and k_u to single indices working on an interval.
standard math Lebesgue monotone differentiation theorem
Used after Lemma 4 to obtain a point u where the monotone boundary map g is differentiable, so the derivative quotients I_n and I_m have limits.
domain assumption Non-compactness of deleted boundary neighborhoods
Asserted in Section 3 without proof: sets (cl U(0,p_n)) without U(0,p_m) are non-compact, which is used to show boundary points map to boundary points.

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Cite this review

Pith. "Pith review of On some modifications of the Niemytzki plane." pith.science (2026). https://pith.science/paper/LOOZ4TRD

@misc{pith2026250601707,
  author       = {Pith},
  title        = {Pith review of: On some modifications of the Niemytzki plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LOOZ4TRD}},
  note         = {Machine review of arXiv:2506.01707}
}

read the original abstract

We present a criterion that compares modifications of the Niemytzki plane. It follows that if usual tangent discs of the Niemytzki plane are replaced by triangles with bounded angles, then the resulting space is not homeomorphic to the former.

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Abuzaid, M

D. Abuzaid, M. Alqahtani, L. Kalantan, New topologies between the usual and Niemytzki, Appl. Gen. Topol. 21 (2020), no. 1, 71–79

work page 2020
[2]

V. A. Chatyrko, On n-dimensional Niemytzki spaces , Filomat 39 (2025), in press

work page 2025
[3]

Engelking, General topology, translated from the Polish by the author, second edition, Sigma Ser

R. Engelking, General topology, translated from the Polish by the author, second edition, Sigma Ser. Pure Math., 6, Heldermann Verlag, Berlin, 1989, viii+529 pp

work page 1989
[4]

Hattori, Order and topological structures of posets of the formal balls on metric spaces, Mem

Y. Hattori, Order and topological structures of posets of the formal balls on metric spaces, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci.43 (2010), 13–26

work page 2010
[5]

Kulesza, Results on spaces between the Sorgenfrey and usual topologies on R, Topology Appl

J. Kulesza, Results on spaces between the Sorgenfrey and usual topologies on R, Topology Appl. 231 (2017), 266–275

work page 2017
[6]

E. E. Moise, Geometric Topology in Dimensions 2 and 3 , Springer Sci- ence+Business Media, LLC, New York 1977

work page 1977
[7]

L. A. Steen, J. A. Seebach, Jr., Counterexamples in topology . Reprint of the second (1978) edition. Dover Publications, Inc., Mineola, NY, 1995. xii+244 pp

work page 1978

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Abuzaid, M

D. Abuzaid, M. Alqahtani, L. Kalantan, New topologies between the usual and Niemytzki, Appl. Gen. Topol. 21 (2020), no. 1, 71–79

work page 2020

[2] [2]

V. A. Chatyrko, On n-dimensional Niemytzki spaces , Filomat 39 (2025), in press

work page 2025

[3] [3]

Engelking, General topology, translated from the Polish by the author, second edition, Sigma Ser

R. Engelking, General topology, translated from the Polish by the author, second edition, Sigma Ser. Pure Math., 6, Heldermann Verlag, Berlin, 1989, viii+529 pp

work page 1989

[4] [4]

Hattori, Order and topological structures of posets of the formal balls on metric spaces, Mem

Y. Hattori, Order and topological structures of posets of the formal balls on metric spaces, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci.43 (2010), 13–26

work page 2010

[5] [5]

Kulesza, Results on spaces between the Sorgenfrey and usual topologies on R, Topology Appl

J. Kulesza, Results on spaces between the Sorgenfrey and usual topologies on R, Topology Appl. 231 (2017), 266–275

work page 2017

[6] [6]

E. E. Moise, Geometric Topology in Dimensions 2 and 3 , Springer Sci- ence+Business Media, LLC, New York 1977

work page 1977

[7] [7]

L. A. Steen, J. A. Seebach, Jr., Counterexamples in topology . Reprint of the second (1978) edition. Dover Publications, Inc., Mineola, NY, 1995. xii+244 pp

work page 1978