Hyperspaces with the Hausdorff Metric and Uniform ANR's
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Abstract
For a metric space X =(X, d), let \Cld_H(X) be the space of all nonempty closed sets in X with the topology induced by the Hausdorff extended metric: d_H(A, B) =max\bigg{\sup<SUB>x∈ B</SUB>d(x, A), \ \sup<SUB>x∈ A</SUB>d(x, B)\bigg} ∈ [0, ∞]. On each component of \Cld_H(X), d_H is a metric (i.e., d_H(A, B) < ∞). In this paper, we give a condition on X such that each component of \Cld_H(X) is a uniform AR (in the sense of E.Michael). For a totally bounded metric space X, in order that \Cld_H(X) is a uniform ANR, a necessary and sufficient condition is also given. Moreover, we discuss the subspace \Dis_H(X) of \Cld_H(X) consisting of all discrete sets in X and give a condition on X such that each component of \Dis_H(X) is a uniform AR and \Dis_H(X) is homotopy dense in \Cld_H(X).
Journal
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- Journal of the Mathematical Society of Japan
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Journal of the Mathematical Society of Japan 57(2), 523-535, 2005-04-01
The Mathematical Society of Japan