Regions below of continuous maps on compact spaces

For a compact metric space X and L=[0, 1] or L=[0, 1], let ↓ C(X, L) denote the family of all regions below of continuous maps from X to L endowed with the topology induced by the Hausdorff metric of the metric space X×[0, 1]. In the present paper, the following result is proved: ↓ C(X, L) is homeomorphic to cο if and only if the set of all isolated points of X is not dense in X where c₀={x_ϵ(-1,1)^∞: lim_n→∞x_n=0}.