A continuum whose hyperspace of regular subcontinua is homeomorphic to the irrationals
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…
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