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

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 semicontinuous 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 δσ -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}.