Gap, Excess and Bornological Convergence

Abstract

Let Open image in new window be the nonempty subsets of a metric space 〈X, d〉. Some classical convergences in Open image in new window - such as convergence in Hausdorff distance, Attouch-Wets convergence and Wijsman convergence - have been shown to be compatible with the weak topology on Open image in new window 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.