Hausdorff hyperspaces of R-m and their dense subspaces

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概要

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• Let Bd(H)(R-m) be the hyperspace of nonempty bounded closed subsets of Euclidean space R-m endowed with the Hausdorff metric. It is well known that Bd(H)(R-m) is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of BdH(R-m) of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space l(2). For each 0 <= 1 < m, letnu(m)(k) = {x=(x(i))(i=1)(m) is an element of R-m : x(i) is an element of R\Q except for at most k many i},where nu(2k+1)(k) is the k-dimensional Nobeling space and nu(m)(0) = (R\Q)(m). It is also proved that the spaces Bd(H)(nu(1)(0)) and Bd(H)(nu(m)(k)), 0 <= k < m - 1, are homeomorphic to l(2). Moreover, we investigate the hyperspace Cld(H)(R) of all nonempty closed subsets of the real line R with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component H of CldH(R) is homeomorphic to the Hilbert space l(2) (2(N0)) of weight 2(N0) in case where H (sic) R, [0, infinity), (-infinity, 0].

• 2008-01-01

著者

• Kubis Wieslaw

  Instytut Matematyki Akademia Swietokrzyska

• Sakai Katsuro

  Institute Of Mathematics University Of Tsukuba

関連論文

• Hausdorff hyperspaces of R-m and their dense subspaces
• The completions of metric ANR's and homotopy dense subsets
• Hyperspaces with the Hausdorff Metric and Uniform ANR's
• Small subdivisions of simplicial complexes with the metric topology
• On topologies of triangulated infinite-dimensional manifolds
• The completions of metric ANR's and homotopy dense subsets

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