HAUSDORFF HYPERSPACES OF R m AND THEIR DENSE SUBSPACES

Let BdH(R ) be the hyperspace of nonempty bounded closed subsets of Euclidean space R endowed with the Hausdorff metric. It is well known that BdH(R ) is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of BdH(R ) 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 6 1 < m, let ν m k = {x = (xi) m i=1 ∈ R m : xi ∈ R \ Q except for at most k many i}, where ν k is the k-dimensional Nöbeling space and ν m 0 = (R \Q) . It is also proved that the spaces BdH(ν 1 0) and BdH(ν m k ), 0 6 k < m−1, are homeomorphic to l2. Moreover, we investigate the hyperspace CldH(R) of all nonempty closed subsets of the real line R with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component H of CldH(R) is homeomorphic to the Hilbert space l2(2 0) of weight 20 in case where H 6∋ R, [0,∞), (−∞, 0].