Metrics that generate the same hyperspace convergence

Costantini, Camillo; Levi, Sandro; Ziemińska, Jolanta

doi:10.1007/bf01027689

Cited by 22 publications

(7 citation statements)

References 8 publications

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“…In §2 we devise and prove a necessary and sufficient condition for an upper gap topology G + S ,d to be finer than (or equal to) another upper gap topology G + T ,ρ , where d, ρ are metrics on a set X not a priori assumed equivalent and S , T are collections of nonempty subsets of X (see Theorem 2.2). It is to be observed that this result generalizes, in a quite faithful way, an analogous result for the upper Wijsman topology [19,Theorem 5 ], which could now be deduced as a corollary. Combining this necessary and sufficent condition with its converse, we obtain a characterization of coincidence of G + S ,d with G + T ,ρ .…”

Section: Introductionsupporting

confidence: 74%

“…Cornet [16] first suggested a unified program of topologies on nonempty closed subsets by viewing such sets as sitting in C(X, R) equipped with topologies of uniform convergence on various bornologies, identifying each nonempty closed set A with its distance functional d(•, A). For the topology of uniform convergence on X one gets the classical Hausdorff pseudometric topology (see, e.g., [4,26]); for the topology of uniform convergence on finite subsets, one gets nothing more than the topology of pointwise convergence, and the resulting topology is called the Wijsman topology, subsequently studied extensively by the authors and their associates [5,12,17,18,19,21,24]; for the topology of uniform convergence on bounded sets, one gets the Attouch-Wets topology [1,4,26,27,28], which plays a fundamental role in convex analysis, as it is stable with respect to duality in arbitrary normed linear spaces, as shown by Beer [2]. Both Hausdorff metric convergence and Attouch-Wets convergence are special cases of so-called bornological convergence as introduced by Lechicki, Levi and Spakowski [25] and studied by the present authors and their associates in subsequent papers [7,10,11].…”

Section: Introductionmentioning

confidence: 99%

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Bornological Convergence and Shields

2011

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In this article we study gap topologies on the subsets of a metric space (X, d) induced by a general family S of nonempty subsets of X. Given two families and two metrics not assumed to be equivalent, we give a necessary and sufficient condition for one induced upper gap topology to be contained in the other. This condition is also necessary and sufficent for containment of the two-sided gap topologies under the mild assumption that the generating families contain the singletons. Coincidence of upper gap topologies in the most important special cases is attractively reflected in the underlying structure of (X, d). First and second countability of upper gap topologies is also characterized. This approach generalizes and unifies results in [12] and [19] and gives rise to a noticeable family of subsets that lie between the totally bounded and the bounded subsets of X.

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Section: Introductionsupporting

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Bornological Convergence and Shields

2011

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“…Unlike lower Wijsman convergence, upper Wijsman convergence does depend on the metric; necessary and sufficient conditions for two metrics to yield the same upper Wijsman convergence (and thus determine the same Wijsman topology) are presented in [14].…”

Section: Lemma 2 Let S Be a Family Of Nonempty Subsets Of A Metric Spmentioning

confidence: 99%

Gap, Excess and Bornological Convergence

Beer

Levi

2008

Set-Valued Anal

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Let P 0 (X) be the nonempty subsets of a metric space X, d . Some classical convergences in P 0 (X) -such as convergence in Hausdorff distance, Attouch-Wets convergence and Wijsman convergence -have been shown to be compatible with the weak topology on P 0 (X) induced by all gap and excess functionals with fixed left argument ranging in some bornology. Here we consider an arbitrary ideal of subsets of X and compare the gap and excess topology so generated with the corresponding convergence defined in terms of truncations by elements of the ideal.

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“…(cf. [3], [15] and [1] Page 38) Let (X, τ ) a Tychonoff space, U ∈ Π(τ ) and D, E ⊂ X. We say that D is strictly U-included in E (D ⊂⊂ U E) iff there exist a finite set F ⊂ D and entourages U , U ∈ U with U composably contained in U such that…”

Section: Preliminariesmentioning

confidence: 99%

Bombay hypertopologies

Maio

Meccariello

Naimpally³

2003

Appl. Gen. Topol.

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<p>Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.</p>

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Metrics that generate the same hyperspace convergence

Cited by 22 publications

References 8 publications

Bornological Convergence and Shields

Bornological Convergence and Shields

Gap, Excess and Bornological Convergence

Bombay hypertopologies

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