On the Fell topology

Beer, Gerald

doi:10.1007/bf01039292

Cited by 34 publications

(19 citation statements)

References 33 publications

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“…The hit-or-miss topology τ f (see [28,29]) (also known as H-topology [9], Fell topology [4,5,21], Choquet-Matheron topology [26], or weak Vietoris topology [32]) on F is generated by the subbase…”

Section: Preliminariesmentioning

confidence: 99%

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

Li¹,

Zhao²,

Wang³

et al. 2017

J. Nonlinear Sci. Appl.

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Section: Preliminariesmentioning

confidence: 99%

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

Li¹,

Zhao²,

Wang³

et al. 2017

J. Nonlinear Sci. Appl.

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“…Let Y be a locally compact space, V a compatible uniformity and ∆ 2 = K(Y ). Then the corresponding ∆ 2 -AW filter V ∆2 on 2 Y is a uniformity (see [4] or [5]) and it will be denoted with U(F ). Moreover, if 2 Y is equipped with the Fell topology τ 2 (F ), it is known that U(F ) is compatible with τ 2 (F ) (see [4] and [10]).…”

Section: Some Special Cases Of δ Arementioning

confidence: 99%

Graph topologies on closed multifunctions

Maio

Meccariello

Naimpally³

2003

Appl. Gen. Topol.

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Dedicated by the first two authors to Professor S. Naimpally on the occasion of his 70 th birthday. Abstract.In this paper we study function space topologies on closed multifunctions, i.e. closed relations on X × Y using various hypertopologies. The hypertopologies are in essence, graph topologies i.e topologies on functions considered as graphs which are subsets of X × Y . We also study several topologies, including one that is derived from the Attouch-Wets filter on the range. We state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions.2000 AMS Classification: 54B20, 54C25, 54C35, 54C60, 54E05, 54E15.

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“…We quote [En] and [Be1] for the basic notions. One of the most important and well-studied hyperspace topologies on CL(X) is the Fell topology [At], [Be1], [Be2], [Fe], [Fl], [Po]. The Fell topology can be considered a classical one, as it has found numerous applications in different fields of mathematics ( [Ma], [At]).…”

mentioning

confidence: 99%

“…By a result in [HL] (CL(X), τ F ) is first countable if and only if X is first countable, hereditarily separable, and each open set is hemicompact (see also [Be2]). The hemicompactness of X together with the first countability of X imply that X is locally compact and Lindelöf.…”

mentioning

confidence: 99%

Normality and paracompactness of the Fell topology

Holá

Levi

Pelant

1999

Proc. Amer. Math. Soc.

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Abstract. Let X be a Hausdorff topological space and CL(X) the hyperspace of all closed nonempty subsets of X. We show that the Fell topology on CL(X) is normal if and only if the space X is Lindelöf and locally compact. For the Fell topology normality, paracompactness and Lindelöfness are equivalent.Throughout the paper all spaces are assumed to be Hausdorff. By X we always denote a space, while CL(X) (resp. K(X)) is the set of all nonempty closed (compact) subsets of X. We quote [En] To describe this topology, we need to introduce some notation. For E a subset of X, we associate the following subsets of CL(X):The Fell topology τ F on CL(X) has as a subbase all sets of the form V − , where V is an open subset of X plus all sets of the form (K c ) + , where K ∈ K(X) and K c is the complement of K. In locally compact spaces, convergence with respect to the Fell topology is Kuratowski convergence of nets of sets.If compact subsets in the above definition are replaced by closed sets, we obtain the stronger Vietoris topology, also called the finite topology [Mi]. The normality of the Vietoris topology on CL(X) is equivalent to the compactness of X as was shown by Veličko in [Ve]. We refer also to Keesling's deep study of normality of the Vietoris topology [Ke1], [Ke2]. The regularity and Hausdorffness of the Fell topology was studied by Poppe [Po] and the complete regularity by Beer and Tamaki in [BT]. We can summarize here these results as follows:

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On the Fell topology

Cited by 34 publications

References 33 publications

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

Graph topologies on closed multifunctions

Normality and paracompactness of the Fell topology

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