1993
DOI: 10.1090/s0002-9947-1993-1094552-x
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Weak topologies for the closed subsets of a metrizable space

Abstract: 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 interp… Show more

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Cited by 45 publications
(9 citation statements)
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“…Let ∆ 1 , ∆ 2 ⊂ CL(X) be such that ∆ 1 contains the singletons. Then τ * coincides with the weak hypertopology τ weak generated by gap and excess functionals (see [2], [9], [20]). …”
Section: Hyperspace Topologiesmentioning
confidence: 99%
“…Let ∆ 1 , ∆ 2 ⊂ CL(X) be such that ∆ 1 contains the singletons. Then τ * coincides with the weak hypertopology τ weak generated by gap and excess functionals (see [2], [9], [20]). …”
Section: Hyperspace Topologiesmentioning
confidence: 99%
“…Figure 2) is semi-locally connected (see 2.5), yet (C(X), τ HM ) is not T 2 . This can be shown in a similar way to Example (1). In fact, the subcontinua For the remainder of the paper the symbol H(X) will denote any of the hyperspaces defined in the previous section.…”
Section: Examplesmentioning
confidence: 66%
“…Il est prouvé dans [9,10,14] que la slice topologie définie sur C (X) est la topologie la moins fine rendant continues toutes les fonctions…”
Section: ++unclassified
“…Aussi, le développement considérable desétudes sur les convergences variationnelles qui constituent un outil puissant et efficace pour l'étude de la stabilité et l'approximation d'une large classe de problèmes variationnels, a permis de dégager plusieurs importantes notions de convergences, dont l'une est la slice convergence qui est une extension naturelle de la convergence au sens de Mosco dans le cas non réflexif. Cette convergence aété introduite formellement pour la première fois dans [43] puisétudiée intensivement par divers auteurs [3,9,10,14,22,23,35]. Elle est en général plus fine que la convergence au sens de Mosco et coïncide avec cette dernière si et seulement si l'espace considéré est réflexif [11].…”
Section: Introductionunclassified