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 $X$, let $D(X)$ be the hyperspace of subcontinua of $X$ equal to the closure of their interior, with the Hausdorff metric. Camargo, Ordo\~nez and Ram\'irez asked whether $D(X)$ can be homeomorphic to the rationals or to the irrationals. We answer the latter question affirmatively. We construct a continuum $X\subseteq\ell^2$ 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 $G_\delta$ subset of a Cantor cube. We then show that no point of $D(X)$ has a compact neighborhood. The Alexandrov--Urysohn characterization yields $D(X)\cong\N^\N$, and hence $D(X)$ 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