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hexagon:2610.00570v1

math.GN — General Topology

A continuum whose hyperspace of regular subcontinua is homeomorphic to the irrationals

Contributed by Lucas Almeida Almeida

For a metric continuum XX, let D(X)D(X) be the hyperspace of subcontinua of XX equal to the closure of their interior, with the Hausdorff metric. Camargo, Ordo\~nez and Ram\'irez asked whether D(X)D(X) can be homeomorphic to the rationals or to the irrationals. We answer the latter question affirmatively. We construct a continuum X⊆ℓ2X\subseteq\ell^2 as the closure of a countable tree of scaled copies of an indecomposable plane continuum. Two types of piece alternate: blocks with dense interior, and spines with empty interior carrying densely attached blocks. We characterize all regular subcontinua by nonempty convex sets of pieces satisfying a density condition, and prove that this coding is a homeomorphism onto a GδG_\delta subset of a Cantor cube. We then show that no point of D(X)D(X) has a compact neighborhood. The Alexandrov--Urysohn characterization yields D(X)≅NND(X)\cong\N^\N, and hence D(X)D(X) is homeomorphic to the irrationals. We also explain why this construction does not extend to the rational case, which remains open. Github repository: https://github.com/LucasAlves8Sp/A-CONTINUUM-WHOSE-HYPERSPACE-OF-REGULAR-SUBCONTINUA-IS-HOMEOMORPHIC-TO-THE-IRRATIONALS

Primarily AI-generated textHuman understanding: all partsAI-assisted mathematicsBaire SpaceContinuumLean 4hyperspacesindecomposable continuumirrational numbersregular subcontinuum
Oct 27, 2026PDF ↗View work →
hexagon:2610.00546v1

math.GN — General Topology

ALL FINITE CARDINALITIES OF THE HYPERSPACE OF REGULAR SUBCONTINUA

Contributed by Lucas Almeida Almeida

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.

Primarily AI-generated textHuman understanding: all partsAI-assisted mathematicsContinuumLean 4enumeration of connected setshyperspacesindecomposable continuuminverse problemregular subcontinuu
Oct 26, 2026PDF ↗View work →