Hausdorff hyperspaces of R^m and their dense subspaces

Abstract

Let Bd_H(R^m) be the hyperspace of nonempty bounded closed subsets of Euclidean space R^m endowed with the Hausdorff metric. It is well known that Bd_H(R^m) is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of Bd_H(R^m) 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. For each 0≤1<m, let
ν^m_k = {x = (x_i)^m_i=1 ∈R^m : x_i ∈R\\setminusQ except for at most k many i},
where ν^2k+1_k is the k-dimensional Nöbeling space and ν^m₀=(R\\setminusQ)^m. It is also proved that the spaces Bd_H(ν¹₀) and Bd_H(ν^m_k), 0≤k<m−1, are homeomorphic to l₂. Moreover, we investigate the hyperspace Cld_H(R) of all nonempty closed subsets of the real line R with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component \\mathcal{H} of Cld_H(R) is homeomorphic to the Hilbert space l₂(2^ℵ₀) of weight 2^ℵ₀ in case where \\mathcal{H}\\ ot\\ iR,[0,∞),(−∞,0].