A pair of spaces of upper semi-continuous maps and continuous maps

For a Tychonoff space X, we use ↓ USC ( X ) and ↓ C ( X ) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I = [ 0 , 1 ] , respectively. In this paper, we consider the spaces ↓ USC ( X ) and ↓ C ( X ) 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 prove that ↓ USC ( X ) is homeomorphic (≈) to the Hilbert cube Q = [ − 1 , 1 ] ω if and only if X is an infinite compact metric space. And we shall prove that ( ↓ USC ( X ) , ↓ C ( X ) ) ≈ ( Q , c 0 ) , where c 0 = { ( x n ) ∈ Q : lim n → ∞ x n = 0 } , if and only if ↓ C ( X ) ≈ c 0 if and only if X is a compact metric space and the set of isolated points is not dense in X.