math.GN — General Topology

On the structure of countable hyperspaces of regular subcontinua

Contributed by Lucas Almeida Almeida

Let $X$ be a metric continuum and let $\mathcal D(X)$ be its hyperspace of regular subcontinua, that is, of the subcontinua of $X$ which are the closure of their interior. We prove that if $\mathcal D(X)$ is countable, then the interiors of its minimal elements are dense in $X$, and every regular subcontinuum is determined by the minimal elements it contains. We further establish that $\mathcal D(X)$ is a $G_\delta$ subset of the hyperspace $C(X)$ of subcontinua whenever $X$ admits a family of regular subcontinua whose interiors have dense union and which satisfies an absorption property. Consequently, every countable $\mathcal D(X)$ is completely metrizable and scattered. In particular, $\mathcal D(X)$ is never homeomorphic to the rational numbers, which answers a question of Ramírez Angarita and of Camargo, Ordoñez and Ramírez. The theorems of the paper are formalized in Lean 4.