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 ∅) 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]ω be the Hilbert cube, B(Q)=Q⧹(−1,1)ω (the pseudo-boundary of Q) and Qf={(xi)i∈N∈Q∣xi=0 except for finitely many i∈N}. In this paper, we prove that Cld∗F(X) is homeomorphic 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)≈B(Q). In case X is strongly countable-dimensional, this is also equivalent to Fin∗F(X)≈Qf.