Hyperspaces of separable Banach spaces with the Wijsman topology

Abstract

Abstract. Let X be a separable metric space. By CldW (X), we de-note 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 Hilbert space `2 and FinW (X) ≈ BddW (X) ≈ `2 × `f2, where `f2 = {(xi)i∈N ∈ `2 | 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 suf-ficient condition under which FinW (X) is homotopy dense in CldW (X). 1