Topology structures of the families of gray images

Abstract

denote all regions below of continuous maps from X to L. For an infinite compact metric space X,↓USC(X,I) with Vietoris topology is homeomorphic to Hilbert cube Q and ↓C(X,I) is its subspace, where Q=[-1, 1]∞. ↓USC(X, I) could be regarded as a mathematical model of all gray images. In the present paper, the following result is proved: ↓USC(X, [0,1)) is homeomorphic to Q\{(0)}. Therefore the topological structure of ↓C(X, [0,1)) is also clear. Keywor ds: Upper semi-continuous maps. Continuous maps. Vietoris topology. Regions below of maps. Hilbert cube. 1.1 Intr oduction For a Tychonoff space X, the hyperspace Cld(X) is the set consisting of all nonempty closed subsets of X endowed with the Vietoris topology which is generated by the subbase {U − , U