The hyperspace of the regions below of continuous maps is homeomorphic to [formula omitted

Abstract

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 G δ σ -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 , c 0 ) , where Q = [ − 1 , 1 ] ω is the Hilbert cube and c 0 = { ( x n ) ∈ Q : lim n → ∞ x n = 0 } .