Hyperspaces with the Attouch-Wets topology homeomorphic to $\ell _2$

R.I.Voytsitskyy

It is shown that the hyperspace of all nonempty closed subsets $\mathrm{Cld}_{AW}(X)$ of a separable metric space $(X,d)$ endowed with the Attouch-Wets topology is homeomorphic to $\ell _2$ if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologically complete and not locally compact at infinity.

1. T. Banakh, M. Kurihara, K. Sakai, Hyperspaces of normed linear spaces with Attouch-Wets topology, Set-Valued Anal. 11 (2003), 21--36.

2. G. Beer, Topologies on closed and closed convex sets, MIA 268, Dordrecht: Kluwer Acad. Publ., 1993.

3. T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, VNTL Publishers, Lviv, 1996.

4. T. Banakh and R. Voytsitskyy, Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts, Topology Appl. (to appear).

5. T. Banakh and R. Voytsitskyy, Characterizing metric spaces whose hyperspaces are homeomorphic to $\ell_2$, preprint.

6. Doug Curtis and Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic to $\aleph_0$-dimensional linear metric spaces, Topol. Appl. 19 (1985), 251-260.

7. C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc. 123 (1995), 2569--2574.

8. D.W.~Curtis, R.M.~Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19--38.

9. D.W. Curtis, Boundary sets in the Hilbert cube, Topology Appl. 20 (1985), 201--221.

10. W. Kubi\'s, K. Sakai, M. Yaguchi Hyperspaces of separable Banach space with the Wijsman topology, Topology Appl. 148 (2005), 7--32.

11. J.D. Lawson Topological semilattices with small semilattices, J.~London Math. Soc. (2) 1 (1969) 719--724.

12. K. Sakai and Z. Yang, Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube, Topology Appl. 127 (2002), 331--342.

	Hyperspaces with the Attouch-Wets topology homeomorphic to $\ell _2$
Author	R.I.Voytsitskyy voytsitski@mail.lviv.ua Department of Mathematics, Ivan Franko Lviv National,University,
Abstract	It is shown that the hyperspace of all nonempty closed subsets $\mathrm{Cld}_{AW}(X)$ of a separable metric space $(X,d)$ endowed with the Attouch-Wets topology is homeomorphic to $\ell _2$ if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologically complete and not locally compact at infinity.
Keywords	hyperspace, Attouch-Wets topology, homeomorphism
DOI	doi:10.30970/ms.29.2.207-214
Reference	1. T. Banakh, M. Kurihara, K. Sakai, Hyperspaces of normed linear spaces with Attouch-Wets topology, Set-Valued Anal. 11 (2003), 21--36. 2. G. Beer, Topologies on closed and closed convex sets, MIA 268, Dordrecht: Kluwer Acad. Publ., 1993. 3. T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, VNTL Publishers, Lviv, 1996. 4. T. Banakh and R. Voytsitskyy, Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts, Topology Appl. (to appear). 5. T. Banakh and R. Voytsitskyy, Characterizing metric spaces whose hyperspaces are homeomorphic to $\ell_2$, preprint. 6. Doug Curtis and Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic to $\aleph_0$-dimensional linear metric spaces, Topol. Appl. 19 (1985), 251-260. 7. C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc. 123 (1995), 2569--2574. 8. D.W.~Curtis, R.M.~Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19--38. 9. D.W. Curtis, Boundary sets in the Hilbert cube, Topology Appl. 20 (1985), 201--221. 10. W. Kubi\'s, K. Sakai, M. Yaguchi Hyperspaces of separable Banach space with the Wijsman topology, Topology Appl. 148 (2005), 7--32. 11. J.D. Lawson Topological semilattices with small semilattices, J.~London Math. Soc. (2) 1 (1969) 719--724. 12. K. Sakai and Z. Yang, Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube, Topology Appl. 127 (2002), 331--342.
Pages	207-214
Volume	29
Issue	2
Year	2008
Journal	Matematychni Studii
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