All Finite Cardinalities of the Hyperspace of Regular Subcontinua

For a metric continuum X, let 𝒟(X) be the hyperspace of regular subcontinua of X: the subcontinua A with A = cl int A. Ordoñez asked which finite numbers occur as |𝒟(X)| (Topology Appl. 234 (2018), Problem 4.14). Camargo, Ordoñez and Ramírez excluded 2, 4, 5, 8 and 9 and asked whether every k ≥ 256 occurs (Appl. Gen. Topol. 25 (2024), Question 3.21). We prove that a positive integer k equals |𝒟(X)| for some metric continuum X if and only if k ∉ {2, 4, 5, 8, 9, 16}. The finite cardinalities of 𝒟(X) are exactly the numbers of vertex sets inducing connected subgraphs of finite connected graphs. Every finite connected graph is realized by copies of the Knaster continuum glued at points. We determine the values of this graph invariant without computer assistance. A short argument excludes 16, and an explicit recursive family of graphs realizes every k ≥ 17 with the least possible number of vertices. Consequently, for k ≥ 32 the least number of minimal regular subcontinua of a continuum X with |𝒟(X)| = k is ⌊log₂ k⌋ + 1. Supplementary material. The GitHub repository holds version 2.3 of the paper (PDF and LaTeX source), together with the following:https://github.com/LucasAlves8Sp/ALL-FINITE-CARDINALITIES-OF-THE-HYPERSPACE-OF-REGULAR-SUBCONTINUA A Lean 4 / Mathlib formalization (Lean 4.35.0, Mathlib commit 6015dbb) that proves Theorem A with no hypotheses, using only Lean’s standard axioms. It was also checked with leanchecker and the external type checker nanoda. Python and C programs that check every explicit finite graph count, with their outputs. CHECKS.md maps each computational statement in the paper to its check. The companion note Small continua with finitely many regular subcontinua: illustrated examples. It draws six explicit continua with |𝒟(X)| = 1, 3, 6, 7, 11 and 17 and lists all their regular subcontinua:https://github.com/LucasAlves8Sp/ALL-FINITE-CARDINALITIES-OF-THE-HYPERSPACE-OF-REGULAR-SUBCONTINUA/blob/main/supplementary-note/illustrated-examples.pdf The AI disclosure and the full first prompt given to the model. Status. Preprint, not yet peer reviewed. No independent expert has verified the proofs yet. AI disclosure. The work was done mainly by Claude Opus 5.5 (Anthropic), directed and revised by the author, with ChatGPT-Astra (OpenAI) used for review. See AI_DISCLOSURE.md in the repository.