For any bounded open interval X in the Euclidean space E 1 , let ↓USC(X) and ↓C(X) be the families of all which is not homeomorphic to (s, c 0). Hence this paper figures out the topological structure of the pair (↓USC(X), ↓C(X)).
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
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