Distance functional and suprema of hyperspace topologies

Beer, Gerald; Lechicki, Alojzy; Levi, Sandro; Naimpally, S. A.

doi:10.1007/bf01760016

Cited by 81 publications

(38 citation statements)

References 18 publications

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“…The Wijsman topology is not only useful in applications but also valuable as the building block of many hypertopologies ( [3]). It was defined in terms of convergence viz.…”

Section: The Wijsman Topologymentioning

confidence: 99%

“…Moreover, Wijsman topologies are the building blocks of many other topologies e.g., (a) the metric proximal topology is the sup of all Wijsman topologies induced by uniformly equivalent metrics, and (b) the Vietoris topology is the sup of all Wijsman topologies induced by topologically equivalent metrics. ( [3]) In addition, the Wijsman topologies are rather intriguing since there are examples of two uniformly equivalent metrics giving non equivalent Wijsman topologies and two non uniformly equivalent metrics, giving equivalent Wijsman topologies! Attempts to topologize the Wijsman topology led to the discovery of (a) the ball topology ( [1]) and (b) the proximal ball topology ( [4]) which were only partially successful.…”

Section: Introductionmentioning

confidence: 99%

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All hypertopologies are hit-and-miss

Naimpally

2002

Appl. Gen. Topol.

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Abstract. We solve a long standing problem by showing that all known hypertopologies are hit-and-miss. Our solution is not merely of theoretical importance. This representation is useful in the study of comparison of the Hausdorff-Bourbaki or H-B uniform topologies and the Wijsman topologies among themselves and with others. Up to now some of these comparisons needed intricate manipulations. The H-B uniform topologies were the subject of intense activity in the 1960's in connection with the Isbell-Smith problem. We show that they are proximally locally finite topologies from which the solution to the above problem follows easily. It is known that the Wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in"nice" metric spaces including the normed linear spaces. With the introduction of a new far-miss topology we show that the Wijsman topology is hit-andmiss for all metric spaces. From this follows a natural generalization of the Wijsman topology to the hyperspace of any T1 space. Several existing results in the literature are easy consequences of our work.2000 AMS Classification: 54B20, 54E05, 54E15.

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“…The Wijsman topology is not only useful in applications but also valuable as the building block of many hypertopologies ( [3]). It was defined in terms of convergence viz.…”

Section: The Wijsman Topologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

All hypertopologies are hit-and-miss

Naimpally

2002

Appl. Gen. Topol.

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“…(1) ^c lim inf04,), (2) d(A, X) > lim sup d(A,, X) for all X 6 F(E), We mention some references related to W : [Wi,BaP,FrLL,LeL,So2,BLLN,Be6].…”

Section: Classes and Types Elementary Propertiesmentioning

confidence: 99%

Set convergences. An attempt of classification

Sonntag¹,

Zălinescu²

1993

Trans. Amer. Math. Soc.

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Abstract. We endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics. Such structure is completely determined by a class (which is a family of closed sets) and a type (which is a semimetric). Two types are sufficient to define (and classify) almost all convergences known till now. These two types offer the possibility of defining other set convergences.

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“…Without local compactness, the topology fails to be Hausdorff. The topology that works for our purposes is the so-called Wijsman topology, studied extensively over the past twenty years by an international cast of characters [Be2,BLLN,Co,DL,FLL,Hel,He2,LL,Na]. Sequential convergence in this topology was introduced by Wijsman [Wi] in the context of convex analysis.…”

Section: Failure Of Standard Hyperspace Topologies To Be Polishmentioning

confidence: 99%

A Polish topology for the closed subsets of a Polish space

Beer¹

1991

Proc. Amer. Math. Soc.

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Abstract.Let (X, d) be a complete and separable metric space. The Wijsman topology on the nonempty closed subset CL(A") of X is the weakest topology on CL{X) such that for each x in X , the distance functional A -* d{x, A) is continuous on CL{X). We show that this topology is Polish, and that the traditional extension of the topology to include the empty set among the closed sets is also Polish. We also compare the Borel class of a closed valued multifunction with its Borel class when viewed as a single-valued function into Ch{X), equipped with Wijsman topology.

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Distance functional and suprema of hyperspace topologies

Cited by 81 publications

References 18 publications

All hypertopologies are hit-and-miss

All hypertopologies are hit-and-miss

Set convergences. An attempt of classification

A Polish topology for the closed subsets of a Polish space

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