Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube. (Q1868861)

For a space \(X\), let \(\text{Cld}^*_F(X)\) denote the space of all closed subsets of \(X\) endowed with the Fell topology. In this paper the authors prove among other things that for a Hausdorff space \(X\), \(\text{Cld}^*_F(X)\) is homeomorphic to the Hilbert cube \(Q\) if and only if \(X\) is locally compact, locally connected, separable metrizable, and has no compact components. There are also similar results for hyperspaces of compact sets and finite sets, respectively.