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ALL FINITE CARDINALITIES OF THE HYPERSPACE OF REGULAR SUBCONTINUA

Primarily AI-generated textHuman understanding: all partsmath.GN — General Topologymath.CO — Combinatorics

Contributed by Lucas Almeida Almeida ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

For a metric continuum X, let D(X) be the family of subcontinua A of X with A = cl_X int_X A. Ordõnez asked which finite numbers occur as |D(X)|. Camargo, Ordõnez and Ramírez excluded 2, 4, 5, 8, 9 and asked whether every k ≥ 256 occurs. We prove that a positive integer k equals |D(X)| for some metric continuum X if and only if k not in {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. Programs check the explicit finite graph counts, and Theorem A is formally verified in Lean.

Provenance statement

This work arose from an AI-assisted research project investigating the finite cardinalities of the hyperspace of regular subcontinua of a metric continuum. The project addresses a question introduced by Ordoñez and subsequently studied by Camargo, Ordoñez, and Ramírez: for which positive integers n does there exist a metric continuum X such that the family D(X) = {A ⊆ X : A is a subcontinuum and A = Cl_X int_X (A)} has cardinality n? The main result establishes that such a continuum exists if and only if n not in {2, 4, 5, 8, 9, 16}. In particular, this settles the previously posed question of whether every integer n ≥ 256 can occur, proving the stronger statement that every integer n ≥ 17 is realizable. The investigation first establishes a correspondence between finite cardinalities of D(X) and the numbers of nonempty vertex sets inducing connected subgraphs of finite connected graphs. Every finite connected graph is realized by a continuum constructed by gluing copies of the Knaster continuum at points. The realization argument does not assume that D(X) is discrete. The graph-theoretic classification then excludes 16 by a direct argument and realizes every integer at least 17 through an explicit recursive family of graphs, together with finitely many base cases. The construction also yields an optimal bound on the number of minimal regular subcontinua needed for each realizable finite cardinality, with the exceptional value 25 treated separately. AI systems were used extensively for mathematical exploration, the development and refinement of proofs, graph constructions, computational verification, Lean formalization, and auditing. The human contributor directed the research, selected the problem and research objectives, evaluated intermediate results, reviewed the mathematical arguments, and is responsible for the claims and presentation of this submission. The project includes a public research repository containing the manuscript and its LaTeX source, supplementary examples, computational programs and data, verification documentation, and a Lean 4 formalization. The repository reports that the main classification theorem is formally verified in Lean 4, using only Lean’s standard axioms, with no ‘sorry‘, added axioms, or native_decide in the formalization. The computational graph-counting claims are checked separately by reproducible programs. These verification efforts provide machine-checkable evidence for the formalized statements and reproducible checks of the computational results; they should not be understood as independent expert review of the entire mathematical work. Repository and verification materials: https://github.com/LucasAlves8Sp/ALL-FINITE-CARDINALITIES-OF-THE-HYPERSPACE-OF-REGULAR-SUBCONTINUA

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Version history

  1. v1Submitted by Lucas Almeida AlmeidaInitial depositCurrentOct 26, 2026