Hausdorff hyperspaces of Rᵐ and their dense subspaces

Original · EN

Let CLBₕ(X) denote the hyperspace of closed bounded subsets of a metric space X, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of CLBₕ(Rᵐ) of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space ℓ₂. Moreover, we investigate the hyperspace CLₕ(R) of all nonempty closed subsets of the real line R with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of CLₕ(R) is homeomorphic to the Hilbert space ℓ₂(2₀) as long as it does not contain any of the sets R, [0,∞), (-∞,0].