Let Bd
H(
Rm) be the hyperspace of nonempty bounded closed subsets of Euclidean space
Rm endowed with the Hausdorff metric. It is well known that Bd
H(
Rm) is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of Bd
H(
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. For each 0≤1<
m, let
ν
mk = {
x = (
xi)
mi=1 ∈
Rm :
xi ∈
R\\setminus
Q except for at most
k many
i},
where ν
2k+1k is the
k-dimensional Nöbeling space and ν
m0=(
R\\setminus
Q)
m. It is also proved that the spaces Bd
H(ν
10) and Bd
H(ν
mk), 0≤
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 \\mathcal{H} of
CldH(
R) is homeomorphic to the Hilbert space
l2(2
ℵ0) of weight 2
ℵ0 in case where \\mathcal{H}\\
ot\\
i
R,[0,∞),(−∞,0].
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