The finite cardinalities of the hyperspace of regular subcontinua: Ordoñez's Problem 4.14 and Camargo–Ordoñez–Ramírez Question 3.21

A continuum is a nonempty compact connected metric space, and a subcontinuum $A$ of $X$ is regular if $A=\overline{\operatorname{int}_X A}$ . Let $\mathcal D(X)$ be the hyperspace of regular subcontinua of $X$ . Ordoñez (2018, Problem 4.14) asked for which $n\in\mathbb N$ there is a continuum $X$ with $|\mathcal D(X)|=n$ ; Camargo, Ordoñez and Ramírez (2024) repeated it as Question 3.5. They showed $|\mathcal D(X)|\notin\{2,4,5,8,9\}$ , found by a computer search over graphs with at most eight vertices that $16$ does not occur and that every value from $17$ to $255$ does, and asked (Question 3.21) whether every $k\ge256$ occurs. Ramírez's thesis also asked whether $|\mathcal D(X)|=2^n$ is possible for $n>0$ .

Result: Proved(see note)
Status: Resolved
AI contribution: AI co-developed
Method: Construction
Field: Continuum theory; hyperspaces and connected induced subgraphs
Posed by: Ordoñez, Topology Appl. 234 (2018), Problem 4.14; repeated as Question 3.5, with Question 3.21, by Camargo, Ordoñez and Ramírez, Appl. Gen. Topol. 25 (2024)
Year posed: 2018
Years open: 8y
Solved: 2026-10-23
Model: Claude Opus 5.5
Vendor: Anthropic
Collaborators: Lucas Alves de Almeida
Verification: Lean-checked, statement unaudited
Publication: Announced
Significance: 6 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem A: a positive integer $k$ equals $|\mathcal D(X)|$ for some metric continuum $X$ if and only if $k\notin\{2,4,5,8,9,16\}$ . This settles Problem 4.14 (Question 3.5) and answers Question 3.21 positively: every $k\ge17$ occurs. Also, $|\mathcal D(X)|=2^m$ exactly when $m=0$ or $m\ge5$ (Pregunta 3.1.13).

The finite values are exactly the numbers of connected vertex sets of finite connected graphs, each graph realized by Knaster continua glued at points. A recursive family realizes every $k\ge17$ on $\lfloor\log_2 k\rfloor+1$ vertices (six for $k=25$ ), the least possible for $k\ge32$ . Context: Czabarka, Székely and Wagner (2009) had shown that every integer from $90$ on is the number of subtrees of a tree, so with a realization of trees the values $k\ge90$ followed from earlier work, as the paper says; the new parts are the realization for all graphs, the computer-free determination and the least-vertex result.

What the AI did

The paper's AI disclosure says most of the work was done by Claude Opus 5.5, in an iterative process directed by the author. The model carried out most of the exploratory proof development, found the near-cones $N(d,R)$ of Section 5 (the family that fills the bottom $m-1$ values of each dyadic interval $[2^{m-1},2^m-1]$ and drives the induction for every $k\ge17$ ), wrote and ran the checking programs of Section 7, and generated most of the first draft of the paper and of the Lean 4 formalization. The author formulated the problem, wrote the initial research prompt with a proposed plan, and supplied the reduction to graphs through Camargo–Ordoñez–Ramírez Theorem 3.16, the realization by Knaster continua glued at points, the exclusion of $16$ , and the Czabarka–Székely–Wagner subtree theorem as a fallback; the author selected, revised and reviewed the final arguments. ChatGPT-Astra (OpenAI) was used to review the paper, proofs and supplementary material over several revision rounds. The unedited first prompt is published in the repository.

Verification

No independent mathematician has checked the proofs yet. Checked by this site on 3 November 2026, not rebuilt. Questions 3.5 and 3.21 and Remark 3.20 were read in the open-access Camargo-Ordoñez-Ramírez paper (Appl. Gen. Topol. 25 (2024)) and match. The Lean headline theoremA (commit 2751bf0, Lean 4.35.0, Mathlib 6015dbb, 18 modules) was read with its definitions: IsDCard quantifies over compact connected metric spaces, regularSubcontinua is the closed connected sets equal to the closure of their interior, and the count is the ncard of a finite set, so the formal statement is the problem as posed, with no hypotheses; both inputs (CORa Theorem 3.16 and the Knaster gluing) are proved as theorems. A grep of all modules found no sorry, admit, axiom declarations, native_decide, unsafe, opaque or implemented_by. The axiom list and the leanchecker and nanoda runs are the author's log; there is no CI. A separate computation written here over every connected graph with at most 7 vertices found the excluded values up to 40 to be exactly 2, 4, 5, 8, 9, 16, and confirmed the least order floor(log2 k)+1 (6 for k=25) for every k from 17 to 127.

Sources

Submitted by StormyGander827 on 24 Oct 2026

The finite cardinalities of the hyperspace of regular subcontinua: Ordoñez's Problem 4.14 and Camargo–Ordoñez–Ramírez Question 3.21

What was actually shown

What the AI did

Verification

Sources

Discussion