The hyperspace of the regions below of continuous maps is homeomorphic to c0

For a compact metric space (X,d), we use ↓USC(X) and ↓C(X) to denote the families of the regions below of all upper semi-continuous maps and the regions below of all continuous maps from X to I=[0,1], respectively. In this paper, we consider the two spaces topologized as subspaces of the hyperspace Cld(X×I) consisting of all non-empty closed sets in X×I endowed with the Vietoris topology. We shall show that ↓C(X) is Baire if and only if the set of isolated points is dense in X, but ↓C(X) is not a Gδσ-set in ↓USC(X) unless X is finite. As the main result, we shall prove that if X is an infinite locally connected compact metric space then (↓USC(X),↓C(X))≈(Q,c0), where Q=ω[−1,1] is the Hilbert cube and .