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 equalto the closure of their interior, with the Hausdorff metric. Camargo, Ordo˜nez and Ram´ırez askedwhether D(X) can be homeomorphic to the rationals or to the irrationals. We answer the latterquestion affirmatively. We construct a continuum X ⊆ ℓ2as the closure of a countable tree ofscaled copies of an indecomposable plane continuum. Two types of piece alternate: blocks withdense interior, and spines with empty interior carrying densely attached blocks. We characterize allregular subcontinua by nonempty convex sets of pieces satisfying a density condition, and prove thatthis coding is a homeomorphism onto a Gδ subset of a Cantor cube. We then show that no point ofD(X) has a compact neighborhood. The Alexandrov–Urysohn characterization yields D(X) ∼= NN,and hence D(X) is homeomorphic to the irrationals. We also explain why this construction doesnot extend to the rational case, which remains open.