math.GN — General Topology
ALL FINITE CARDINALITIES OF THE HYPERSPACE OF REGULAR SUBCONTINUA
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.