Paths in hyperspaces

Camillo Constantini, Wieslaw Kubís

Abstract

We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.


Keywords

Hyperspace; Wijsman topology; Hausdorff metric; Path-wise connectedness; Absolute retract

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References

H.A. Antosiewicz and A. Cellina, Continuous extensions of multifunctions, Ann. Polon. Math. 34 (1977) 107-111.

T. Banakh, M. Kurihara and K. Sakai, Hyperspaces of normed linear spaces with the Attouch-Wets topology, preprint.

G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers 1993.

K. Borsuk and S. Mazurkiewicz, Sur l'hyperespace d'un continu, C. R. Soc. Sc. Varsovie 24 (1931) 149-152.

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

D. Curtis and Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic to N0-dimensional linear metric spaces, Topology Appl. 19 (1985) 251-260. http://dx.doi.org/10.1016/0166-8641(85)90005-7

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

A. Illanes and S. Nadler, Hyperspaces, Marcel-Dekker, New York 1999.

W. Kubis, K. Sakai and M. Yaguchi, Hyperspaces of separable Banach spaces with the Wijsman topology, preprint.

J.D. Lawson, Topological semilattices with small subsemilattices, J. London Math. Soc. (2) 1 (1969), 719-724. http://dx.doi.org/10.1112/jlms/s2-1.1.719

S. Nadler, Hyperspaces of Sets, Marcel-Dekker 1978.

K. Sakai and Z. Yang, Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube, preprint.

L.E. Ward, Jr., Arcs in hyperspaces which are not compact, Proc. Amer. Math. Soc. 28 (1971) 254-258. http://dx.doi.org/10.1090/S0002-9939-1971-0275376-0

M.M. McWaters, Arcs, semigroups and hyperspaces, Can. J. Math. 20 (1968) 1207-1210. http://dx.doi.org/10.4153/CJM-1968-115-3

M. Wojdys lawski, Retractes absolus et hyperespaces des continus, Fund. Math. 32 (1939) 184-192.

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1. Hausdorff hyperspaces of R m and their dense subspaces
Wiesław KUBIŚ, Katsuro SAKAI
Journal of the Mathematical Society of Japan  vol: 60  issue: 1  year: 2008  
doi: 10.2969/jmsj/06010193



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

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