Infima of hyperspace topologies

Costantini, Camillo; Levi, Sandro; Pelant, Jan

doi:10.1112/s0025579300011360

Cited by 9 publications

(8 citation statements)

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“…We know that topologization of isotone convergences is easier to construct. Indeed, if Q is an isotone convergence on £, the TQ-closed sets are precisely those subsets of E which are closed under Q-limits of nets [8,Lemma 2.1].…”

Section: Basic Notionsmentioning

confidence: 99%

“…A similar result about the supremum does not hold in general, even for two topologies (see Example 7.1). Given a collection {T ; | / E /} of topologies on a set E, it is known (see [8]) that its convergence-infimum and its infimum are related as follows:…”

Section: Proposition the Infimum Of Any Collection Of Sequential Topmentioning

confidence: 99%

“…In many cases investigation has been carried out from the lattice-theoretical standpoint and suprema or infima of topologies are studied (see [3,4,8,16]). Thus [3], [4] and [8] deal with topologies on the hyperspace of a metrizable space X, each of which depends on a particular metric on X; suprema or infima of these topologies, over the collection of all compatible metrics, have the property of being metric-independent.…”

Section: Introductionmentioning

confidence: 99%

“…In § 3 we show that r + = inf{H^| d e M(X)} coincides with the sequential modification of a newly defined topology U + , which is closely related to the upper Kuratowski convergence on ^(X). Moreover, we compare T + with the infimum of all upper Wijsman topologies (which is studied in [8]). Section 4 is devoted to the topology T~ = inf{Hj | d e M(X)}\ we prove that xĩ s the sequential modification of the lower Vietoris topology V~.…”

Section: Introductionmentioning

confidence: 99%

“…Since H^s* r + , we also have r + -convergence, and the proof is complete. Now we establish necessary and sufficient conditions under which # U + = TK + : this is of some relevance because the former topology is inf{Hj"| d e M(X)} (Theorem 3.5) while the latter is inf{W,| d e M(X)} [8,Theorem 6.4].…”

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confidence: 99%

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On The Infimum of the Hausdorff Metric Topologies

Costantini

Vitolo

1995

Proceedings of the London Mathematical Society

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Given a metrizable space X and a compatible metric d, one defines the Hausdorff metric topology H d and the upper and lower Hausdorff topologies corresponding to d, H^ and H^ respectively, on the collection ^(X) of all closed subsets of X.In this paper we consider the infima z, T + and r~, of the topologies H d , HJ and HJ respectively, where d runs over the set M{X) of all compatible metrics on X. These topologies are sequential, that is, they are completely characterized by convergent sequences.In particular, the topologies r + and r~ are investigated in detail: a suitable topology U + is defined which has the same convergent sequences as T + , and the lower Vietoris topology V~ plays a similar role with respect to T~.We show that, in general, the equality T = T + V T " does not hold. We also show that T is a r 2 -topology on ^(X) if and only if X is locally compact.

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Section: Basic Notionsmentioning

confidence: 99%

Section: Proposition the Infimum Of Any Collection Of Sequential Topmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On The Infimum of the Hausdorff Metric Topologies

Costantini

Vitolo

1995

Proceedings of the London Mathematical Society

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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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