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January, 2008 Hausdorff hyperspaces of R m 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 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 Bd H ( 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 2 . For each 0 1 < m , let

ν k m = { x = ( x i ) i = 1 m R m : x i R Q except for at most k many i } ,

where ν k 2 k + 1 is the k -dimensional Nöbeling space and ν 0 m = ( R Q ) m . It is also proved that the spaces Bd H ( ν 0 1 ) and Bd H ( ν k m ) , 0 k < m - 1 , are homeomorphic to 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 Cld H ( R ) is homeomorphic to the Hilbert space 2 ( 2 0 ) of weight 2 0 in case where H R , [ 0 , ) , ( - , 0 ] .

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Wiesław KUBIŚ. Katsuro SAKAI. "Hausdorff hyperspaces of R m 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

Keywords: bounded closed sets , Cantor sets , Euclidean space , Hilbert space , Lebesgue measure zero , Nöbeling space , nowhere dense closed sets , perfect sets , the Hausdorff metric , the Hilbert cube , the hyperspace , the pseudo-interior

Rights: Copyright © 2008 Mathematical Society of Japan

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