## Journal of the Mathematical Society of Japan

### Hausdorff hyperspaces of $\mathbf{R}^{m}$ and their dense subspaces

#### Abstract

Let $\mathrm{Bd}_{H}(\mathbf{R}^{m})$ be the hyperspace of nonempty bounded closed subsets of Euclidean space $\mathbf{R}^{m}$ endowed with the Hausdorff metric. It is well known that $\mathrm{Bd}_{H}(\mathbf{R}^{m})$ is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of $\mathrm{Bd}_{H}(\mathbf{R}^{m})$ 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 $\ell_{2}$. For each $0 \leq 1 < m$, let

$\nu^{m}_{k} = \{x = (x_{i})_{i=1}^{m} \in \mathbf{R}^{m} : x_{i} \in \mathbf{R}\setminus\mathbf{Q}$ except for at most $k$ many $i \}$,

where $\nu^{2k+1}_{k}$ is the $k$-dimensional Nöbeling space and $\nu^{m}_{0} = (\mathbf{R}\setminus\mathbf{Q})^{m}$. It is also proved that the spaces $\mathrm{Bd}_{H}(\nu^{1}_{0})$ and $\mathrm{Bd}_{H}(\nu^{m}_{k})$, $0\leq k, are homeomorphic to $\ell_{2}$. Moreover, we investigate the hyperspace $\mathrm{Cld}_{H}(\mathbf{R})$ of all nonempty closed subsets of the real line $\mathbf{R}$ with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component $\mathcal{H}$ of $\mathrm{Cld}_{H}(\mathbf{R})$ is homeomorphic to the Hilbert space $\ell_{2}(2^{\aleph_{0}})$ of weight $2^{\aleph_{0}}$ in case where $\mathcal{H}$$\mathbf{R}, [0,\infty), (-\infty,0]$.

#### Article information

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

Dates
First available in Project Euclid: 24 March 2008

https://projecteuclid.org/euclid.jmsj/1206367960

Digital Object Identifier
doi:10.2969/jmsj/06010193

Mathematical Reviews number (MathSciNet)
MR2392008

Zentralblatt MATH identifier
1160.54004

#### 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

#### References

• [1] \auR. D. Anderson, D. W. Henderson and J. E. West, Negligible subsets of infinite-dimensional manifolds, \tiCompositio Math., , 21 ((1969),)\spg143–\epg150.
• [2] \auH. A. Antosiewicz and A. Cellina, Continuous extensions of multifunctions, \tiAnn. Polon. Math., , 34 ((1977),)\spg107–\epg111.
• [3] T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, Math. Studies Monog. Ser., 1, VNTL Publishers, Lviv, 1996.
• [4] \auT. Banakh and R. Voytsitskyy, Characterizing metric spaces whose hyperpsaces are absolute neighborhood retracts, \tiTopology Appl., , 154 ((2007),)\spg2009–\epg2025.
• [5] G. Beer, Topologies on Closed and Closed Convex Sets, Math. Appl., 268, Kluwer Academic Publ., Dordrecht, 1993.
• [6] \auT. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, \tiTrans. Amer. Math. Soc., , 154 ((1971),)\spg399–\epg426.
• [7] \auC. Costantini, Every Wijsman topology relative to a Polish space is Polish, \tiProc. Amer. Math. Soc., , 123 ((1995),)\spg2569–\epg2574.
• [8] \auC. Costantini and W. Kubiś, Paths in hyperspaces, \tiAppl. Gen. Topology, , 4 ((2003),)\spg377–\epg390.
• [9] \auD. W. Curtis, Hyperspaces of noncompact metric spaces, \tiCompositio Math., , 40 ((1980),)\spg139–\epg152.
• [10] \auD. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, \tiFund. Math., , 101 ((1978),)\spg19–\epg38.
• [11] \auD. W. Curtis and Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic to $\aleph_{0}$-dimensional linear metric spaces, \tiTopology Appl., , 19 ((1985),)\spg251–\epg260.
• [12] \auW. H. Cutler, Negligible subsets of infinite-dimensional Fréchet manifolds, \tiProc. Amer. Math. Soc., , 23 ((1969),)\spg668–\epg675.
• [13] A. Illanes and S. B. Nadler, Jr., Hyperspaces, Fundamentals and Recent Advances, Pure Applied Math., 216, Marcell Dekker, Inc., New York, 1999.
• [14] T. Jech, Set Theory, The third millennium edition, revised and expanded, Springer Monog. in Math., Springer-Verlag, Berlin, 2003.
• [15] A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Math., 156, Springer-Verlag, New York, 1995.
• [16] \auW. Kubiś, K. Sakai and M. Yaguchi, Hyperspaces of separable Banach spaces with the Wijsman topology, \tiTopology Appl., , 148 ((2005),)\spg7–\epg32.
• [17] \auM. Kurihara, K. Sakai and M. Yaguchi, Hyperspaces with the Hausdorff metric and uniform ANRs, \tiJ. Math. Soc. Japan, , 57 ((2005),)\spg523–\epg535.
• [18] \auJ. D. Lawson, Topological semilattices with small subsemilattices, \tiJ. London Math. Soc., , 1 ((1969),)\spg719–\epg724.
• [19] J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library, 43, Elsevier Science Publisher B.V., Amsterdam, 1989.
• [20] \auJ. Saint Raymond, La structure borélienne d'Effros est-elle standard?, \tiFund. Math., , 100 ((1978),)\spg201–\epg210.
• [21] \auK. Sakai, On hyperspaces of polyhedra, \tiProc. Amer. Math. Soc., , 110 ((1990),)\spg1089–\epg1097.
• [22] \auH. Torunczyk, Characterizing Hilbert space topology, \tiFund. Math., , 111 ((1981),)\spg247–\epg262.
• [23] \auH. Torunczyk, A correction of two papers concerning Hilbert manifolds, \tiFund. Math., , 125 ((1985),)\spg89–\epg93.