{"id":"hexagon/2610.00546","pdf":"https://hexagonmath.org/pdf/2610.00546v1","url":"https://hexagonmath.org/abs/2610.00546v1","tags":["math.CO","msc:54B20","AI-assisted mathematics","Continuum","Lean 4","enumeration of connected sets","hyperspaces","indecomposable continuum","inverse problem","regular subcontinuu"],"score":0,"title":"ALL FINITE CARDINALITIES OF THE HYPERSPACE OF REGULAR SUBCONTINUA","domain":"math.GN","origin":"ai-generated","source":"hexagon","authors":["Lucas Almeida Almeida"],"reviews":0,"subject":"math.GN","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}