A pair of spaces of upper semi-continuous maps and continuous maps |
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Authors: | Zhongqiang Yang Xiaoe Zhou |
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Affiliation: | a Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, PR China b School of Business, Shantou University, Shantou, Guangdong, 515063, PR China |
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Abstract: | For a Tychonoff space X, we use ↓USC(X) and ↓C(X) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I=[0,1], respectively. In this paper, we consider the spaces ↓USC(X) and ↓C(X) 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 prove that ↓USC(X) is homeomorphic (≈) to the Hilbert cube Q=ω[−1,1] if and only if X is an infinite compact metric space. And we shall prove that (↓USC(X),↓C(X))≈(Q,c0), where , if and only if ↓C(X)≈c0 if and only if X is a compact metric space and the set of isolated points is not dense in X. |
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Keywords: | 54B20 54D05 54E45 57N20 |
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