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

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]. 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.