The hyperspace of the regions below continuous maps with the fell topology

Abstract

For a Tychonoff space X, we use ↓USC F (X) and ↓C F (X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0, 1] with the subspace topologies of the hyperspace Cld F (X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology. In this paper, we shall show that there exists a homeomorphism h: ↓USC F (X) → Q = [−1, 1] ω such that h(↓C F (X)) = c 0 = {(x n ) ∈ Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.