Hyperspaces with the Hausdorff Metric and Uniform ANR's

• KURIHARA Masayuki

• SAKAI Katsuro

  Institute of Mathematics University of Tsukuba

• YAGUCHI Masato

  Institute of Mathematics University of Tsukuba

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_{x∈ B}d(x, A), \ \sup_{x∈ A}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

• Journal of the Mathematical Society of Japan

  Journal of the Mathematical Society of Japan 57 (2), 523-535, 2005

  The Mathematical Society of Japan

Keywords

• Hyperspace of closed sets
• Hausdorff metric
• uniform AR
• uniform ANR
• uniformly locally C^*-connected
• almost convex
• <i>C</i>-connected
• Lawson semilattice

Details

• CRID: 1390282680091937024
• NII Article ID: 10016420078
• NII Book ID: AA0070177X
• DOI: 10.2969/jmsj/1158242069
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• MRID: 2123243
• NDL BIB ID: 7304305
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I7304305
• Text Lang: en
• Data Source:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
• Abstract License Flag: Disallowed