Hyperspaces of Normed Linear Spaces with the Attouch–Wets Topology

Set-Valued Analysis - 2003
Taras Banakh1, Masayuki Kurihara2, Katsuro Sakai2
1Department of Mathematics, Lviv National University, Lviv, Ukraine
2Institute of Mathematics, University of Tsukuba, Tsukuba, Japan

Tóm tắt

Let Cld AW (X) be the hyperspace of nonempty closed subsets of a normed linear space X with the Attouch–Wets topology. It is shown that the space Cld AW (X) and its various subspaces are AR's. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then Cld AW (X) is homeomorphic to a Hilbert space with weight 2 w(X).

Từ khóa


Tài liệu tham khảo

Beer, G.: Topologies on Closed and Closed Convex Sets, MIA 268, Kluwer Acad. Publ., Dordrecht, 1993.

Bessaga, C. and Pełczy´nski, A.: Selected Topics in Infinite-Dimensional Topology, MM 58, Polish Sci. Publishers, Warsaw, 1975.

Curtis, D.W.: Hyperspaces homeomorphic to Hilbert space, Proc. Amer. Math. Soc. 75 (1979), 126-130.

Curtis, D. W.: Hyperspaces of noncompact metric spaces, Compositio Math. 40 (1980), 139-152.

Curtis, D. and Nguyen To Nhu: Hyperspaces of finite subsets which are homeomorphic to 0-dimensional linear metric spaces, Topology Appl. 19 (1985), 251-260.

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

Cutler, W. H.: Negligible subsets of infinite-dimensional Fréchet manifolds, Proc. Amer. Math. Soc. 23 (1969), 668-675.

Hu, S.-T.: Theory of Retracts, Wayne State Univ. Press, Detroit, 1965.

Illanes, A. and Nadler, S. B., Jr.: Hyperspaces, Fundamentals and Recent Advances, Pure and Appl. Math. 216, Marcel Dekker Inc., New York, 1999.

Jameson, G. J. O.: Topology and Normed Spaces, Chapman and Hall, London, 1974.

Kurihara, M.: Hyperspaces of metric spaces, Master Thesis, University of Tsukuba, 2000.

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

Lowen, R. and Sioen,M.: TheWijsman and Attouch-Wets topologies on hyperspaces revisited, Topology Appl. 70 (1996), 179-197.

Megginson, R. E.: An Introduction to Banach Space Theory, GTM 183, Springer, New York Inc., New York, 1998.

van Mill, J.: Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library 43, Elsevier, Amsterdam, 1989.

Nguyen To Nhu: Hyperspaces of compact sets in metric linear spaces, Topology Appl. 22 (1986), 109-122.

Sakai, K. and Uehara, S.: Spaces of upper semicontinuous multi-valued functions on complete metric spaces, Fund. Math. 160 (1999), 199-218.

Sakai, K. and Yang, Z.: Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube, Topology and Appl., in press.

Tašmetov, U.: On the connectedness of hyperspaces, Dokl. Akad. Nauk SSSR 215 (1974), 286-288 (Russian); English transl., Soviet Math. Dokl. 15 (1974), 502-504.

ń, H.: Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247-262.

ń, H.: A correction of two papers concerning Hilbert manifolds, Fund. Math. 125 (1985), 89-93.

Wojdysławski, M.: Rétractes absoulus et hyperespaces des continus, Fund. Math. 32 (1939), 184-192.

Thông tin
Thông tin xuất bản

Set-Valued Analysis

Thông tin tác giả