The hyperspaces $HS(p,X)$

Florencio Corona-V\'azquez; Javier S\'anchez Mart\'inez; Russell Aar\'on Qui\~nones-Estrella

REVIEW 4 major objections 7 minor 34 references

The hyperspaces $HS(p,X)$

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

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Pith's one-line read The new quotient hyperspace $HS(p,X)=C(X)/C(p,X)$ is a continuum whose local and global properties track those of $X$, and in finite graphs it detects the circle.

desk verdict A useful first study of a new hyperspace quotient, but Theorem 9.1's proof gap undercuts the colocality section; the rest is mostly standard and worth a refereed revision. read the letter →

arxiv 1908.06200 v2 pith:YTLM46EO submitted 2019-08-16 math.GN

Florencio Corona-Vázquez , Russell Aarón Quiñones-Estrella , Javier Sánchez Martínez This is my paper · ORCID

classification math.GN MSC 54B0554B2054F65

keywords continuahyperspacesquotientspacesC(pX)unicoherenceproperty(b)finitegraphsKelley

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 introduces a new quotient of the hyperspace of subcontinua: fix $p\in X$ and collapse the entire family $C(p,X)$ of continua that contain $p$ to a single point, producing $HS(p,X)=C(X)/C(p,X)$. The payoff is a continuum that reflects the topology of $X$, and in some cases sharpens it. The paper establishes that $HS(p,X)$ is always unicoherent and has property (b), that under the Kelley property two such quotients at distinct points are locally connected exactly when $X$ is locally connected, and that for a finite graph with ordinary point $p$, $C(X)$ is homeomorphic to $HS(p,X)$ if and only if $X$ is a simple closed curve. A final characterization uses embeddability in the plane to single out arcs and simple closed curves among Kelley continua.

What carries the argument

The carrying object is the quotient map $\pi_p^X:C(X)\to HS(p,X)$, where $C(X)$ is the hyperspace of subcontinua of $X$ with the Hausdorff metric and all members of $C(p,X)$ are identified with one point. The map is monotone, meaning each fiber is a continuum, so known theorems on monotone images transfer unicoherence and property (b) from $C(X)$ to $HS(p,X)$. For graph comparisons the paper uses familiar geometric models of $C(X)$ (a 2-cell for an arc, a cone for a fan, an n-cell with attached 2-cells for an n-od) together with a dimension formula for hyperspaces of finite graphs; order arcs move a continuum $A$ to a larger one inside $C(X)$ while keeping track of its distance from $C(p,X)$.

What would settle it

Take $X=[0,1]$, $p=1/2$, an open interval $W$ around $p$ with $\delta=\sup_{w\in W}|w-p|$, and some $A\in C(X)-N(\delta,C(p,X))$; follow the order arc from a singleton $\{a\}$ with $a\in A$ to $A$. If any intermediate continuum on that arc lies inside the $\delta$-neighborhood of $C(p,X)$ within $C(X)$, the connectivity assertion behind Theorem 9.1 fails.

Watch

Extended reading notes

Core claim

The central claim is that $HS(p,X)$ is a continuum whose topological properties are inherited from, and in the right circumstances characterize, the base continuum $X$. The paper proves that $HS(p,X)$ is uniformly pathwise connected, unicoherent, and has property (b) for every $p$, and that $C(p,X)$ collapses to a cut point exactly when $p$ cuts $X$. For a finite graph and an ordinary point $p$, it proves $C(X)\cong HS(p,X)$ if and only if $X$ is a simple closed curve. Under the Kelley property, $X$ is locally connected if and only if $HS(p,X)$ and $HS(q,X)$ are locally connected for two distinct points $p$ and $q$; and if $X$ is a locally connected continuum without free arcs, each $HS(p,X)$ is homeomorphic to the Hilbert cube. A further theorem says that, under Kelley, planar embeddability of the two quotients at two colocally connected points forces $X$ to be an arc or a simple closed curve.

Load-bearing premise

The load-bearing premise is that in the proof of Theorem 9.1 the complement $C(X)-N(\delta,C(p,X))$ is connected by order arcs from far singletons; the written argument's distance inequality appears reversed, and the later aposyndesis and arc-versus-circle characterizations (Corollaries 9.2, 9.3, Theorem 9.4) depend on this unproved step.

Editorial extensions

If this is right

Since $HS(p,X)$ is a monotone quotient of $C(X)$, every hyperspace property preserved by monotone images, including unicoherence and property (b), holds for $HS(p,X)$ for every point $p$.
For finite graphs and ordinary $p$, the homeomorphism $C(X)\cong HS(p,X)$ is a topological test for being a simple closed curve, so the quotient separates the circle from every other finite graph at ordinary points.
Under the Kelley property, local connectedness of $X$ can be checked by looking at only two quotient spaces, $HS(p,X)$ and $HS(q,X)$, at any two distinct points.
If $X$ is a locally connected continuum without free arcs, the collapse changes nothing up to homeomorphism: both $C(X)$ and $HS(p,X)$ are Hilbert cubes.
A Kelley continuum whose quotients at two distinct points are planar-embeddable and colocally connected at those points must be an arc or a simple closed curve.

Reading between the lines

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

The monotone-image argument suggests the same construction works for collapsing any closed connected family of continua in $C(X)$, not only those through a single point; quotienting by larger families would give a hierarchy of hyperspace invariants.
The Hilbert-cube result for locally connected continua without free arcs indicates $HS(p,X)$ carries no new information in that class, so the real discriminative power sits in one-dimensional graphs, where the dimension formula makes the change at ramification points explicit.
The tree question left open in the paper, which end-point collapses preserve $C(X)$, can be attacked by comparing the number of components of the 2-dimensional layers $U_2(C(X))$ and $U_2(HS(p,X))$, the invariant used in the proof of Theorem 5.12.
Theorem 7.1's two-point criterion raises a question the paper does not ask: for a fixed $X$, which points $p$ make $HS(p,X)$ locally connected? Under Kelley the answer is uniform, but outside that class the set of such points could be a recognizable subspace of $X$.

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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

Short version: a useful first study of a new hyperspace quotient, with most sections resting on standard material; but Theorem 9.1 has a genuine, load-bearing proof gap and the colocality results should not be accepted as they stand.

What's genuinely new is the object itself: HS(p,X)=C(X)/C(p,X), a natural relative of the Nadler/Macías suspensions. The paper correctly observes that the quotient map is monotone and leverages standard hyperspace theorems to get dimension formulas, unicoherence, property (b), local-connectedness transfer under Kelley, and aposyndesis-type results. The concrete examples for the arc, circle, n-od and the yarn example help the reader see the quotient. The proof of Theorem 5.12 using component counts of U2 is a nice touch. For Sections 4–8, the mathematics looks coherent and the citation pattern is reasonable; leaning on the authors' unpublished preprint for Corollary 5.7 is a soft spot but not a serious one.

The problem is Theorem 9.1. The step meant to show C(X)−N(δ,C(p,X)) is connected does not work: from A in that complement you only get H(A,{p}) ≥ δ, which gives a∈A with d(a,p) ≥ δ, not ≤ δ as printed; and that does not force a outside V. More importantly, no argument shows an order arc from {a} to A stays in the complement. Hausdorff distance is not monotone under inclusion, so intermediate subcontinua can be much closer to C(p,X) than A is. On top of that, the notation N(δ,C(p,X)) is ambiguous in the hyperspace, and the proof never verifies that this neighbourhood maps into the prescribed open set W. Because Corollaries 9.2 and 9.3 and Theorem 9.4 depend on 9.1, the colocality section is currently unproved. This looks repairable, but it is a real gap, not a cosmetic one.

My own verdict: the paper deserves a serious referee, but only with an explicit request to reconstruct Theorem 9.1. If that argument is fixed, the rest is publishable in a standard topology venue. For hyperspace and continuum-theory researchers it is a useful new tool. I would not cite the current version in my own work until the gap is closed.

Referee Report

4 major / 7 minor

Summary. The paper defines, for a continuum X and a point p∈X, the quotient hyperspace HS(p,X)=C(X)/C(p,X), and studies how its topological properties reflect those of X. After a section of concrete examples (arc, simple closed curve, n-od, and graph examples), the paper proves dimension bounds and a dimension comparison for finite graphs (Section 4), basic properties including functoriality, detection of cut points, and a characterization of when HS(p,X) is homeomorphic to C(X) for finite graphs (Section 5), unicoherence and property (b) (Section 6), local connectedness under the Kelley property (Section 7), aposyndesis results (Section 8), and colocality results (Section 9), including a characterization of arcs and simple closed curves under the Kelley property.

Significance. HS(p,X) is a natural quotient of C(X), and the paper's examples and functorial framework should make it a useful object for continuum theory. Sections 4 through 8 are mostly clean applications of standard hyperspace and monotone quotient theorems, and the dimension and cut-point results are informative. The intended colocality theorem (Theorem 9.1) and its corollaries are the main new contribution, but the proof as written has substantial gaps; since the later characterization (Theorem 9.4) depends on those results, the paper's central claim is not yet established.

major comments (4)

[Section 9, Theorem 9.1, second case] The displayed inference 'H(A,{p}) ≥ δ, then there exists a∈A such that d(a,p) ≤ δ' is reversed. The correct conclusion is d(a,p) ≥ δ. Since the chosen open set in X, called W in the proof, lies in B_{ε/2}(p), we have δ ≤ ε/2, so d(a,p) ≥ δ does not imply a∉V. The claimed point in F1(X−V) is therefore not obtained.
[Section 9, Theorem 9.1, second case] Even if a∉V were obtained, the order arc α from {a} to A is not shown to stay in C(X)−N(δ,C(p,X)). Hausdorff distance is not monotone with respect to inclusion, so a subcontinuum of A can be closer to some B∈C(p,X) than A itself. The assertion that α([0,1]) is a connected subset of C(X)−N(δ,C(p,X)) is unjustified.
[Section 9, Theorem 9.1, second case] No argument shows that N(δ,C(p,X)) ⊂ (π^X_p)^{-1}(W). The ε-ball around the singleton {p} is not a neighbourhood of the whole fiber C(p,X), and δ is chosen from the open set V in X without reference to the quotient map. Consequently π(N(δ,C(p,X))) is not shown to be the desired neighbourhood of C^X_p inside W.
[Section 9, Corollaries 9.2 and 9.3 and Theorem 9.4] Because these results are derived directly from Theorem 9.1, the gaps in the proof of Theorem 9.1 propagate to all of them. Theorem 9.1 is the sole source for the aposyndesis and finite-aposyndesis corollaries and for the 2⇒1 direction of Theorem 9.4, so these statements are presently unsupported.

minor comments (7)

[Section 4, Corollary 4.5] The proof cites 'Corollary 4.1', but no Corollary 4.1 exists; Lemma 4.1 is presumably meant.
[Section 9, proof of Theorem 9.1] The symbol W is used both for an open subset of HS(p,X) and for an open subset of X; the latter should be renamed V to avoid confusion.
[Section 9, proof of Theorem 9.1] The sentence 'In order to prove that HS(p,X) is locally connected in C^X_p' should say 'colocally connected'; local connectedness at that point is not the goal of the argument.
[Section 7, Theorem 7.1, and Section 10, Question 10.4] The name 'Kelly' should be 'Kelley' in both places.
[Section 3, Example 3.4] The space is defined as Y but the surrounding text and diagram use X; a single symbol should be used consistently.
[Section 5, Corollary 5.7] The proof depends entirely on the unpublished preprint [10, Theorem 4.14], which is neither stated nor proved here. Since Corollary 5.7 is a stated theorem of the paper, this missing support should be addressed even though the corollary is not used in later sections.
[Section 5, Lemma 5.9] The expression |π0(U2(HS(p,X))| is missing a closing parenthesis; it should be |π0(U2(HS(p,X)))|.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity; one minor self-citation in Corollary 5.7 is load-bearing only for that side result.

uniqueness imported from authors [Section 5, Corollary 5.7 (proof)]
"Corollary 5.7. Let X and Y two trees, p ∈ X and q ∈ Y . If C(p, X ) is homeomorphic to C(q, Y ) then HS (p, X ) is homeomorphic to HS (q, Y ). Proof. By [10, Theorem 4.14], we have that there exists a homeomorphism h : X → Y sending p to q, now apply Corollary 5.6 to conclude."

The corollary's entire proof is an appeal to [10, Theorem 4.14], an unpublished preprint by the same authors that supplies a uniqueness theorem for C(p,X) in trees. Thus a claim about HS(p,X) is made to rest on the authors' own prior unverified result. This is not circular in the narrow sense of assuming the target conclusion, but it is a self-citation whose load-bearing content is not independently established in the present paper. It affects only Corollary 5.7; the main theorems (5.12, 6.1, 6.2, 7.1, 9.4) do not depend on it.

full rationale

I walked the paper's derivation chains. The main results are derived from standard published theorems: dimension facts from Hurewicz–Wallman and Martinez de la Vega, unicoherence from the known unicoherence of C(X) and monotone quotients, property (b) from Nadler and Kuratowski, local connectedness from standard hyperspace results, and the arc/simple-closed-curve characterization from internal lemmas plus standard graph hyperspace facts. There is no fitting of parameters, no quantity is renamed as a prediction, and no known result is repackaged as new by definition. The Skeptic's concern about Theorem 9.1 is a real internal proof gap: the displayed inequality appears reversed and the order-arc assertion is not justified, but a proof gap is a correctness problem, not circularity, so it does not raise the circularity score. The only circularity-adjacent item is Corollary 5.7, which imports a uniqueness theorem from the authors' own unpublished preprint [10]; that self-citation is load-bearing only for that corollary and does not support the central claims. I therefore assign a score of 2.

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

The paper's central claims rest on standard results in continuum theory and hyperspace theory. The only non-standard, paper-specific premise is the unpublished preprint [10] used in Corollary 5.7. No free parameters are fitted and no new physical entities are postulated.

assumptions (4)

domain assumption The quotient map pi_X^p from C(X) to HS(p,X) is monotone, and monotone images of unicoherent continua and of property (b) continua retain those properties.
Used in Section 6 (Theorems 6.1 and 6.2) to prove HS(p,X) is unicoherent and has property (b); citing Whyburn [34, 1.21] and Kuratowski [22, Theorem 2].
standard math C(X) is unicoherent and has property (b) for every continuum X.
Cited from Illanes-Nadler [19, 19.8] and Nadler [27, Theorem 3].
standard math For X locally connected without free arcs, C(X) is homeomorphic to the Hilbert cube Q, and removing a contractible AR subset of Q leaves a complement homeomorphic to Q minus a point.
Used in Theorem 7.2 via Curtis-Schori [11, 4.1], Eberhart [13, Theorem 2], and Chapman [6, 25.2].
ad hoc to paper The uniqueness theorem for C(p,X) in the class of trees stated in the authors' unpublished preprint [10, Theorem 4.14] is correct.
No proof is given in this paper; the result is from the authors' own unreviewed preprint, and Corollary 5.7 depends on it.

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Pith. "Pith review of The hyperspaces $HS(p,X)$." pith.science (2026). https://pith.science/paper/YTLM46EO

@misc{pith2026190806200,
  author       = {Pith},
  title        = {Pith review of: The hyperspaces $HS(p,X)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YTLM46EO}},
  note         = {Machine review of arXiv:1908.06200}
}

abstract

Let $X$ be a continuum and let $C(X)$ denote the hyperspace of subcontinua of $X$, endowed with the Hausdorff metric. For $p\in X$, define the hyperspace $C(p,X)=\{A\in C(X):p\in A\}$ as a subspace of $C(X)$. In this paper we introduced the quotient space $HS(p,X)=C(X)/C(p,X)$. We present some general properties of $HS(p,X)$ and we study the relationship between the continuum $X$ and the hyperspaces $C(X)$ and $HS(p,X)$.

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Reference graph

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Pith tools

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

[1] [1]

Barragan, On the n-fold symmetric product suspension of a conti- nuum, Topology Appl., 157 (2010), 597–604

F. Barragan, On the n-fold symmetric product suspension of a conti- nuum, Topology Appl., 157 (2010), 597–604

work page 2010

[2] [2]

Bennett, Aposyndetic properties of unicoherent continua , Paciﬁc J

D.E. Bennett, Aposyndetic properties of unicoherent continua , Paciﬁc J. Math. 37 (1971) 585–589

work page 1971

[3] [3]

Borsuk, Theory of Shape , Lectures, Fall, 1970, in: Lecture Notes Series, vol

K. Borsuk, Theory of Shape , Lectures, Fall, 1970, in: Lecture Notes Series, vol. 28, Aarhus Universitet, Matematisk Institut, 1973

work page 1970

[4] [4]

Casta˜ neda–Alvarado and J

E. Casta˜ neda–Alvarado and J. S´ anchez–Mart ´ ınez,On the unicoherence of Fn(X) and SF n m(X) of continua, Topology Proc., 42 (2013), 309–326

work page 2013

[5] [5]

R. E. Chandler, Hausdorﬀ Compactiﬁcations , in: Lecture Notes Pure Appl. Math., vol. 23, Marcel Dekker, New York, 1976

work page 1976

[6] [6]

Chapman, Lectures on Hilbert Cube Manifolds, in: CBMS Regional Conf

T.A. Chapman, Lectures on Hilbert Cube Manifolds, in: CBMS Regional Conf. Ser. Math., vol. 28, American Mathematical Society, Providence, RI, 1976

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[7] [7]

J. J. Charatonik, On fans , Dissertationes Math. 54, (1967) 1–37

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[8] [8]

J. J. Charatonik and A. Illanes, Local connectedness in hyperspaces , Rocky Mountain J. of Math., 36 (3) (2006) 811–856. 21

work page 2006