arxiv: 2604.12517 · v1 · submitted 2026-04-14 · 🧮 math.GN

Recognition: unknown

The converse to Borsuk's result on fans fails

Benjamin Vejnar

Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:59 UTC · model grok-4.3

classification 🧮 math.GN

keywords fanBorsuk theoremhereditarily unicoherent continuumbranching pointarc-wise connectedone-dimensional continuumcounterexample

0 comments

The pith

Not every union of arcs intersecting at one point is a fan

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

The paper shows that Borsuk's result on fans has no converse. Borsuk proved that every fan, an arc-wise connected hereditarily unicoherent continuum with exactly one branching point, can be written as the union of arcs where any two intersect only at that point. The authors give a more general result that produces a one-dimensional continuum satisfying the union property without being a fan. A reader would care because this means the arc-union description does not fully capture the topological conditions that define fans.

Core claim

Borsuk showed that every fan is a one-dimensional continuum expressible as the union of a family of arcs such that the intersection of any two is exactly the branching point. The paper proves that the converse does not hold by providing a more general result: there exists a continuum that admits such a union representation yet fails to be arc-wise connected and hereditarily unicoherent with exactly one branching point.

What carries the argument

A one-dimensional continuum constructed as the union of arcs all intersecting at a single common point but failing to satisfy hereditary unicoherence.

If this is right

The arc-union property alone does not force a continuum to be a fan.
Hereditary unicoherence and arc-wise connectedness with one branching point are independent of the union representation in at least one direction.
Classification of fans requires the full definition rather than the union property by itself.
Other continua may share the arc-union structure without qualifying as fans.

Where Pith is reading between the lines

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

The same union property might appear in non-fan continua embedded in the plane, suggesting the need for embedding-dependent conditions.
Researchers could test whether minimal modifications to the counterexample produce borderline cases between fans and non-fans.
Analogous converses for related objects such as dendrites or dendroids may also fail for similar reasons.

Load-bearing premise

The constructed continuum meets the union-of-arcs condition while violating hereditary unicoherence or the single-branching-point requirement under standard definitions.

What would settle it

Explicit verification that the constructed continuum contains a subcontinuum disconnected by removal of a point other than the common intersection point would show it is not hereditarily unicoherent and hence not a fan.

Figures

Figures reproduced from arXiv: 2604.12517 by Benjamin Vejnar.

**Figure 1.** Figure 1: The quotient map q : F → Y . The Cantor fan F Let F be the cone over C, i.e., F is the quotient of C ×I where C × {1} is pushed to a point, which will be called t. The continuum F is usually called a Cantor fan with top point t. For c ∈ C and a < b ∈ I, we denote by Ic[a, b] the set {c} × [a, b]; analogously, we use Ic[a, b) for intervals open from the right side, i.e., {c} × [a, b), etc. We will also shor… view at source ↗

**Figure 2.** Figure 2: The quotient map q : Y → Z. Additionally, identify the arc psq(I1[1/2, 1]) with the arc As⌢0 ⊆ Zs⌢0 by a homeomorphism, say hs⌢0, of these arcs that maps ts to ts⌢0. Finally, let X be the quotient of W and denote by u : W → X the quotient map. The family of arcs L We describe the family of arcs L = S {Ls : s ∈ Λ ∪ {∅}}. We will proceed in different ways for s = ∅ and s ∈ Λ. The family L∅ of arcs in Y Let … view at source ↗

**Figure 3.** Figure 3: The domain of the quotient map u : W → X. Also u(Zs) ∩ u(Zs ′) = {t} unless s is a prolongation of s ′ by one element or vise versa. Hence we may suppose that K ∈ Ls and L ∈ Ls⌢n with s ∈ Λ and n ∈ ω. Since u(Zs) ∩ u(Zs⌢n) = u(Bs⌢n). However, the only K ∈ Ls that intersects the arc u(Bs⌢n) in a point other than t, intersects it in the opposite end-point. That end-point is, however, not used by any L ∈ Ls⌢n… view at source ↗

read the original abstract

A fan is an arc-wise connected hereditarily unicoherent continuum with exactly one branching point. By a result of Borsuk, every fan is a 1-dimensional continuum that can be expressed as the union of a family of arcs, each pair of which intersects in the branching point. In this paper, we prove that the converse does not hold by providing a more general result.

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

Vejnar shows the converse to Borsuk's theorem on fans fails by giving a counterexample via a more general result.

read the letter

This paper establishes that the converse to Borsuk's result on fans fails. Vejnar shows this by providing a more general result that includes a counterexample: a 1-dimensional continuum that is the union of arcs with pairwise intersections at a single point, but which is not arc-wise connected and hereditarily unicoherent with exactly one ramification point. The paper does well by directly addressing and resolving this question in continuum theory. It is concise and sticks to the mathematical point, giving credit to Borsuk's original work. The negative result is valuable because it clarifies the boundary of the characterization. The soft spots are minor and mostly about presentation. The abstract is very brief, so the construction of the counterexample needs to be carefully checked in the full text to confirm it meets all the conditions for the union while failing the fan properties. No internal contradictions or circularity are evident, and the definitions align with standard usage in the field. This is a paper for specialists in general topology who work on continua and fans. A reader who knows the literature on Borsuk's theorem would get value from seeing where the converse breaks down. It deserves a serious referee because it is a clean, specific result that adds to the understanding of these spaces. I would recommend sending it for peer review rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The paper claims that the converse to Borsuk's theorem on fans fails. A fan is defined as an arc-wise connected hereditarily unicoherent continuum with exactly one branching point. Borsuk proved that every such fan is 1-dimensional and expressible as the union of a family of arcs whose pairwise intersections occur only at the branching point. The manuscript asserts that this property does not characterize fans, by establishing a more general result that produces a counterexample continuum satisfying the union-of-arcs condition but failing to be a fan.

Significance. If the counterexample is correctly constructed and verified, the result would separate the geometric representation property (union of arcs meeting at one point) from the full fan axioms (arc-wise connectedness plus hereditary unicoherence), refining the classification of 1-dimensional continua in continuum theory. The paper does not supply machine-checked proofs, reproducible code, or explicit falsifiable predictions, but a valid counterexample would be a concrete, checkable contribution to the field.

major comments (1)

[Abstract] Abstract (and entire manuscript as provided): the claim that 'the converse does not hold by providing a more general result' is asserted without any construction of the counterexample continuum, without the statement of the more general result, and without any indication of how the example satisfies the 1-dimensional union-of-arcs condition while violating arc-wise connectedness or hereditary unicoherence. This absence makes it impossible to verify that the example meets the topological requirements or that the intersections are confined to a single point as required.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for identifying the lack of detail in the abstract and the provided manuscript excerpt. We agree that the current abstract is overly terse and does not outline the counterexample or the more general result, which hinders verification. We will revise the abstract and main text accordingly.

read point-by-point responses

Referee: [Abstract] Abstract (and entire manuscript as provided): the claim that 'the converse does not hold by providing a more general result' is asserted without any construction of the counterexample continuum, without the statement of the more general result, and without any indication of how the example satisfies the 1-dimensional union-of-arcs condition while violating arc-wise connectedness or hereditary unicoherence. This absence makes it impossible to verify that the example meets the topological requirements or that the intersections are confined to a single point as required.

Authors: We acknowledge this is a valid criticism of the abstract's brevity. The full manuscript contains the construction of a specific 1-dimensional continuum X that is the union of a family of arcs with pairwise intersections only at a single point p, yet X fails to be arcwise connected (hence not a fan) while satisfying the union-of-arcs property. The more general result is a theorem characterizing when such a union-of-arcs continuum is hereditarily unicoherent. To address the concern, we will expand the abstract to state the more general theorem and briefly describe the counterexample's key properties (including why it violates arcwise connectedness but meets the geometric condition). We will also add explicit verification steps in the main text confirming the intersections are confined to p and that the continuum is 1-dimensional. revision: yes

Circularity Check

0 steps flagged

No circularity; counterexample construction is independent

full rationale

The paper cites Borsuk's external theorem on fans, then constructs an explicit counterexample continuum satisfying the arc-union property but failing to be a fan (i.e., not arc-wise connected hereditarily unicoherent with exactly one ramification point). This is a standard direct disproof via counterexample and does not reduce any claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The derivation chain is self-contained against the stated topological definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard definitions from continuum theory without introducing new parameters or entities.

axioms (1)

domain assumption Standard definitions of continua, arcs, arc-wise connectedness, and hereditarily unicoherent spaces in general topology.
The paper invokes these established concepts to define fans and state Borsuk's result.

pith-pipeline@v0.9.0 · 5338 in / 986 out tokens · 35692 ms · 2026-05-10T13:59:38.116757+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Towards the complete classification of fans

I. Baniˇ c, G. Erceg, A. Illanes, I. Jeli´ c, J. Kennedy, and V. Nall. Towards the complete classification of fans.arXiv e-prints, page arXiv:2504.07267, Apr. 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[2]

K. Borsuk. A theorem on fixed points.Bull. Acad. Polon. Sci. Cl. III., 2:17–20, 1954

1954
[3]

K. Borsuk. A countable broom which cannot be imbedded in the plane.Colloq. Math., 10:233–236, 1963

1963
[4]

J. J. Charatonik. On ramification points in the classical sense.Fund. Math., 51:229–252, 1962/63

1962
[5]

J. J. Charatonik. On fans.Dissertationes Math. (Rozprawy Mat.), 54:39, 1967

1967
[6]

Engelking.Theory of dimensions finite and infinite, volume 10 ofSigma Series in Pure Mathematics

R. Engelking.Theory of dimensions finite and infinite, volume 10 ofSigma Series in Pure Mathematics. Heldermann Verlag, Lemgo, 1995

1995
[7]

J. E. Hanson. Compact metrizable 1-dimensional space that is a disjoint union of arcs. MathOverflow. URL:https://mathoverflow.net/q/507984 (version: 2026-02-10)

2026
[8]

Illanes and P

A. Illanes and P. Krupski. On local fixed or periodic point properties.J. Fixed Point Theory Appl., 17(3):495–505, 2015

2015
[9]

Kuratowski.Topology

K. Kuratowski.Topology. Vol. II. Academic Press, New York-London; Pa´ nstwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, new edition, 1968. Trans- lated from the French by A. Kirkor

1968
[10]

D. S. Lipham. On a local property of fences and fans.arXiv e-prints, page arXiv:2508.03530, Aug. 2025

work page arXiv 2025
[11]

S. B. Nadler, Jr.Continuum theory, volume 158 ofMonographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1992. An introduction

1992
[12]

B. Vejnar. A topological characterization of the Sierpi´ nski triangle.Topology Appl., 159(5):1404–1408, 2012

2012
[13]

R. Winkler. Topological automorphisms of modified Sierpi´ nski gaskets realize arbitrary finite permutation groups.Topology Appl., 101(2):137–142, 2000. 7

2000