  It is shown that the hyperspace of all nonempty closed subsets $\Cld_{AW}(X)$
of a separable metric space $X$ endowed with the Attouch-Wets topology is
homeomorphic to a separable Hilbert space if and only if the completion of $X$
is proper, locally connected and contains no bounded connected component, $X$
is topologically complete and not locally compact at infinity.
