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Camargo-Ordoñez-Ramírez Question 3.26, irrational case: a continuum whose hyperspace of regular subcontinua is homeomorphic to the irrationals

For a metric continuum XX, let D(X)\mathcal D(X) be the hyperspace of regular subcontinua of XX (subcontinua AA with A=int⁡XA‾A=\overline{\operatorname{int}_X A}), with the Hausdorff metric. Camargo, Ordoñez and Ramírez (Appl. Gen. Topol. 25 (2024), Question 3.26; earlier in Ramírez Angarita's 2023 thesis, Pregunta 3.2.14) asked: does there exist a continuum XX such that D(X)\mathcal D(X) is homeomorphic to the rationals Q\mathbb Q or to the irrationals I=R∖Q\mathbb I=\mathbb R\setminus\mathbb Q?

Result
Proved(see note)
Status
Partial result
AI contribution
AI co-developed
Method
Construction
Field
Continuum theory, hyperspaces
Posed by
Camargo, Ordoñez and Ramírez (2024), Question 3.26; earlier Ramírez Angarita (2023), Pregunta 3.2.14
Year posed
2023
Years open
3y
Solved
2026-10
Model
Claude Opus 5.5
Vendor
Anthropic
Collaborators
Lucas Alves de Almeida
Verification
Unreviewed
Publication
Preprint
Significance
7 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Proved (irrational case): there is a continuum X⊆ℓ2X\subseteq\ell^2 with D(X)≅NN≅I\mathcal D(X)\cong\mathbb N^{\mathbb N}\cong\mathbb I. The regular subcontinua are characterized by nonempty convex sets of pieces satisfying a density condition; this coding is a homeomorphism onto a GδG_\delta subset of a Cantor cube, no point has a compact neighborhood, and Alexandrov–Urysohn applies. Not proved (rational case): whether some continuum has D(X)≅Q\mathcal D(X)\cong\mathbb Q remains open. The paper only explains why this method cannot give Q\mathbb Q (it always yields a Polish space, and independent choices give 2ℵ02^{\aleph_0} regular subcontinua near each point) and proves the classification of the nested-chain example with D(L)≅ω2\mathcal D(L)\cong\omega^2; no general obstruction is claimed.

What the AI did

Per the paper's AI disclosure: the author set the task, supplied the literature, gave the directions and feedback at each stage; Claude Opus 5.5 proposed the construction (a null tree of scaled Knaster bucket-handle continua in ℓ2\ell^2 alternating 'blocks' with dense interior and 'spines' with empty interior), wrote the proofs and produced a Lean 4/Mathlib formalization of the combinatorial core. ChatGPT-Astra (OpenAI) was used with the author to check successive versions. The full first prompt is in the repository (prompt/first-prompt.md).

Verification

No independent mathematician has checked this yet. Checked by this site on 3 November 2026, not rebuilt. Question 3.26 was read in the open-access Camargo-Ordoñez-Ramírez paper (Appl. Gen. Topol. 25 (2024)) and the statement matches. The Lean development (commit 0033dd1, Lean 4.35.0 with Mathlib, about 830 lines) was read: a grep found no sorry, admit, axiom declarations, native_decide, unsafe, opaque or implemented_by. Its headline theoremA is conditional. It takes two hypotheses: the Alexandrov-Urysohn characterization of Baire space (correctly stated, Kechris 7.7), and GeometricIdentification, which asserts that some continuum has its hyperspace of regular subcontinua homeomorphic to the code space. The second is the geometric heart of the paper (Theorem 4.6, Proposition 5.2), so Lean checks only the code-space half: a nonempty zero-dimensional Polish space in which every compact set has empty interior. The axiom list and the leanchecker and nanoda runs are the author's; there is no CI and nothing was re-run here. The geometric proof was not checked. The rational case remains open.

Sources

Submitted by StormyGander827 on

Changelog2 changes

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