Gap, Excess and Bornological Convergence

Abstract

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.

Bornological structures on many-valued sets

“…Specifically, much interest is in the research of bornological universes that is triples (X, T , B), where T is a topology and B is a bornology on a set X. General bornological spaces play a key role in recent research of convergence structures on hyperspaces [2], [3], [26], in optimization theory [5] and in the study of topologies on function spaces [4], [29].…”

On the semi-uniformity of bornological convergence

“…where S is an element of S. Properties of these distance functions and the convergence structures they define are extensively studied in [5,6,7,16]. There is (at least) one notable difference between this construction and the construction used to define the ρ-Hausdroff distances.…”

New types of completeness in metric spaces

“…Examples of these properties are the bounded compactness, the uniform local compactness, the cofinal completeness, the strong cofinal completeness, recently introduced by Beer in [6], as well as the so-called UC-ness for metric spaces. The study of all these spaces have shown to be not only interesting by themselves but also in connection with some problems in Convex Analysis, in Optimization Theory and in the setting of Convergence Structures on Hyperspaces (see for instance [7] and references therein).…”