Gap, excess and bornological convergence

Let P₀(X) be the nonempty subsets of a metric space 〈X, d〉,. Some classical convergences in P₀(X) - such as convergence in Hausdorff distance, Attouch-Wets convergence and Wijsman convergence - have been shown to be compatible with the weak topology on P₀(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. © 2008 Springer Science+Business Media B.V.