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January, 2008 Hausdorff hyperspaces of Rm and their dense subspaces
Wiesław KUBIŚ, Katsuro SAKAI
J. Math. Soc. Japan 60(1): 193-217 (January, 2008). DOI: 10.2969/jmsj/06010193

Abstract

Let BdH(Rm) be the hyperspace of nonempty bounded closed subsets of Euclidean space Rm endowed with the Hausdorff metric. It is well known that BdH(Rm) is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of BdH(Rm) 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 2. For each 01<m, let

νkm={x=(xi)i=1mRm:xiRQ except for at most k many i},

where νk2k+1 is the k-dimensional Nöbeling space and ν0m=(RQ)m. It is also proved that the spaces BdH(ν01) and BdH(νkm), 0k<m-1, are homeomorphic to 2. Moreover, we investigate the hyperspace CldH(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 2(20) of weight 20 in case where HR,[0,),(-,0].

Citation

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Wiesław KUBIŚ. Katsuro SAKAI. "Hausdorff hyperspaces of Rm and their dense subspaces." J. Math. Soc. Japan 60 (1) 193 - 217, January, 2008. https://doi.org/10.2969/jmsj/06010193

Information

Published: January, 2008
First available in Project Euclid: 24 March 2008

zbMATH: 1160.54004
MathSciNet: MR2392008
Digital Object Identifier: 10.2969/jmsj/06010193

Subjects:
Primary: 54B20, ‎57N20‎

Rights: Copyright © 2008 Mathematical Society of Japan

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