Hyperspaces of separable Banach spaces with the Wijsman topology

Abstract

Let X be a separable metric space. By Cld W ( X ) , we denote the hyperspace of non-empty closed subsets of X with the Wijsman topology. Let Fin W ( X ) and Bdd W ( X ) be the subspaces of Cld W ( 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 Cld W ( X ) is homeomorphic to (≈) the separable Hilbert space ℓ 2 and Fin W ( X ) ≈ Bdd W ( X ) ≈ ℓ 2 × ℓ 2 f , where ℓ 2 f = { ( x i ) i ∈ N ∈ ℓ 2 | x i = 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 Fin W ( X ) and Cld W ( X ) are ANR's. We also give a sufficient condition under which Fin W ( X ) is homotopy dense in Cld W ( X ) .