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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weak topologies for the closed subsets of a metrizable space


Authors: Gerald Beer and Roberto Lucchetti
Journal: Trans. Amer. Math. Soc. 335 (1993), 805-822
MSC: Primary 54B20; Secondary 54E35
DOI: https://doi.org/10.1090/S0002-9947-1993-1094552-X
MathSciNet review: 1094552
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Abstract: The purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space. It can be shown that all of the standard hyperspace topologies--including the Hausdorff metric topology, the Vietoris topology, the Attouch-Wets topology, the Fell topology, the locally finite topology, and the topology of Mosco convergence--arise as weak topologies generated by families of geometric functionals defined on closed sets. A key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of Hausdorff excess functionals.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1094552-X
Keywords: Hyperspace, gap functional, Hausdorff excess, Hausdorff metric topology, Attouch-Wets topology, Wijsman topology, Vietoris topology, distance functional, weak topologies
Article copyright: © Copyright 1993 American Mathematical Society

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