Hyperspaces of separable Banach spaces with the Wijsman topology

Let X be a separable metric space. By CldW(X), we denote the hyperspace of non-empty closed subsets of X with the Wijsman topology. Let FinW(X) and BddW(X) be the subspaces of CldW(X) consisting of all non-empty finite sets and of all non-empty bounded closed sets, respectively. It is proved that if X is an infinite-dimensional separable Banach space then CldW(X) is homeomorphic to (≈) the separable Hilbert space ℓ2 and FinW(X)≈BddW(X)≈ℓ2×ℓ2f, where ℓ2f={(xi)i∈N∈ℓ2|xi=0 except for finitely many i∈N}. Moreover, we show that if the complement of any finite union of open balls in X has only finitely many path-components, all of which are closed in X, then FinW(X) and CldW(X) are ANR's. We also give a sufficient condition under which FinW(X) is homotopy dense in CldW(X).