Weak topologies for the closed subsets of a metrizable space

Beer, Gerald; Lucchetti, Roberto

doi:10.1090/S0002-9947-1993-1094552-X

Weak topologies for the closed subsets of a metrizable space
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Trans. Amer. Math. Soc. 335 (1993), 805-822 Request permission

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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