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On the structure of countable hyperspaces of regular subcontinua

A mix of human-written and AI-generated textHuman understanding: some partsmath.GN — General Topology

Contributed by Lucas Almeida Almeida ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Let XX be a metric continuum and let D(X)\mathcal D(X) be its hyperspace of regular subcontinua, that is, of the subcontinua of XX which are the closure of their interior. We prove that if D(X)\mathcal D(X) is countable, then the interiors of its minimal elements are dense in XX, and every regular subcontinuum is determined by the minimal elements it contains. We further establish that D(X)\mathcal D(X) is a GδG_\delta subset of the hyperspace C(X)C(X) of subcontinua whenever XX admits a family of regular subcontinua whose interiors have dense union and which satisfies an absorption property. Consequently, every countable D(X)\mathcal D(X) is completely metrizable and scattered. In particular, D(X)\mathcal D(X) is never homeomorphic to the rational numbers, which answers a question of Ramírez Angarita and of Camargo, Ordoñez and Ramírez. The theorems of the paper are formalized in Lean 4.

Provenance statement

Developed with extensive assistance from AI systems: Opus 5.5, Fable 5.2 (Claude Fable 5.2, Anthropic) and ChatGPT-Astra. Fable 5.2 carried out the main research; ChatGPT-Astra rechecked the arguments, compared them with the literature and re-derived the results. The author conducted the investigation, chose the questions to pursue in view of the obstructions met, redirected the search, and edited and checked the text. The manuscript was drafted with AI assistance. Human checking: the author has read, edited and checked the text, but not every proof has been verified line by line by a human, and no independent human referee has examined it. Machine checking: the statements of all numbered theorems, lemmas, propositions and corollaries of Sections 1 and 3-6 and Proposition 7.5 are proved in Lean 4 (code written by Fable 5.2) and pass Lean Comparator in a sandbox; the examples and remarks are not formalized. See the declaration at the end of the paper.

Tools used

Anthropic
Claude FableVersion 5.2
Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion Astra
Lean
LeanVersion 4.35.0-rc2 with Mathlib

References

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  3. Camargo, Javier; Ordoñez, Norberto; Ramírez, Diego. On the hyperspaces of meager and regular continua. Appl. Gen. Topol., vol. 25, no. 2, pp. 385–406. 2024DOI
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  9. Ordoñez, Norberto. The hyperspace of regular subcontinua. Topology Appl., vol. 234, pp. 415–427. 2018DOI
  10. Ordoñez, Norberto. Hyperspaces through regular and meager subcontinua. Topology Appl., vol. 300, pp. Paper No. 107760. 2021DOI
Show all 12 referencesShow fewer references
  1. Ramírez Angarita, Diego Alexánder. Los hiperespacios de subcontinuos regulares y subcontinuos magros. 2023. Available at https://noesis.uis.edu.co/entities/publication/cd4f0fe6-ec75-47d8-8e0a-617149375321. Undergraduate thesis (Matemático), Universidad Industrial de Santander, Bucaramangalink
  2. The mathlib Community. The Lean mathematical library. Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), pp. 367–381. 2020

Version history

  1. v1 · 20 pagesSubmitted by Lucas Almeida AlmeidaInitial depositCurrentNov 11, 2026