Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube

By Cld*(F) (X), we denote the space of all closed sets in a space X (including the empty set 0) with the Fell topology. The subspaces of Cld*(F) (X) consisting of all compact sets and of all finite sets are denoted by Comp*(F) (X) and Fin*(F) (X), respectively. Let Q = [-1, 1](omega) be the Hilbert cube, B (Q) = Q/ (1, 1)(omega) (the pseudo-boundary of Q) and Q(f) = {(x(i))(iis an element ofN) is an element of Q \ x(i) = 0 except for finitely many i is an element of N}. In this paper, we prove that Cld*(F) (X) is homeomorphic to (approximate to) Q if and only if X is a locally compact, locally connected separable metrizable space with no compact components. Moreover, this is equivalent to Comp*(F) (X) approximate to B(Q). In case X is strongly countable-dimensional, this is also equivalent to Fin*(F) (X) approximate to Q(f). (C) 2002 Elsevier Science B.V. All rights reserved.