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] ^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.