Regular Subcontinuum

A regular subcontinuum of a metric continuum is a nonempty compact, connected subset that equals the topological closure of its interior in . The associated hyperspace is

where consists of the nonempty subcontinua of , and both hyperspaces carry the Hausdorff metric (Ordoñez 2018).

Every nondegenerate closed interval in is regular relative to . If is an indecomposable continuum, then because every proper subcontinuum has empty interior. More generally, is compact iff it is finite (Ordoñez 2018, Camargo et al. 2024).

Alves de Almeida (2026b) reported that a positive integer occurs as for some metric continuum iff . These finite cardinalities are also exactly the numbers of nonempty sets of graph vertices whose induced subgraphs are connected, taken in finite connected graphs. Distinct sets of graph vertices are counted separately. The construction realizes each such graph by copies of an indecomposable continuum joined at points.

Alves de Almeida (2026a) separately reported a metric continuum for which

Here denotes homeomorphism, has the product topology of countably many copies of the discrete natural numbers, and has the relative topology inherited from . Thus this hyperspace is homeomorphic to the irrational numbers, also called Baire space in this setting. The construction uses a countable tree of shrinking copies of an indecomposable continuum. The corresponding question for the rational numbers remains open in the cited work.

The author credits Claude Opus 5.5 with substantial construction, proof, computation, and formalization work. As of Oct. 25, 2026, neither paper had received independent expert verification. The finite classification has an author-reported Lean formalization. For the irrational-space construction, Lean verifies properties of a combinatorial code space, while its geometric identification with and the Alexandrov-Urysohn characterization remain explicit hypotheses. The final homeomorphism from to the irrational numbers is not formalized (Alves de Almeida 2026ab).