  Let $CLB_H(X)$ 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 $CLB_H(R^m)$ 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 $\ell_2$.
  Moreover, we investigate the hyperspace $CL_H(R)$ of all nonempty closed
subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We
show that a nonseparable component of $CL_H(R)$ is homeomorphic to the Hilbert
space $\ell_2(2^{\aleph_0})$ as long as it does not contain any of the sets $R,
[0,\infty), (-\infty,0]$.
