Journal of the Mathematical Society of Japan

Hausdorff hyperspaces of R m and their dense subspaces

Wiesław KUBIŚ and Katsuro SAKAI

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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 ] .

Article information

Source
J. Math. Soc. Japan, Volume 60, Number 1 (2008), 193-217.

Dates
First available in Project Euclid: 24 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1206367960

Digital Object Identifier
doi:10.2969/jmsj/06010193

Mathematical Reviews number (MathSciNet)
MR2392008

Zentralblatt MATH identifier
1160.54004

Subjects
Primary: 54B20: Hyperspaces 57N20: Topology of infinite-dimensional manifolds [See also 58Bxx]

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

Citation

KUBIŚ, Wiesław; SAKAI, Katsuro. Hausdorff hyperspaces of $\mathbf{R}^{m}$ and their dense subspaces. J. Math. Soc. Japan 60 (2008), no. 1, 193--217. doi:10.2969/jmsj/06010193. https://projecteuclid.org/euclid.jmsj/1206367960


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