Set Convergences via bornology

This paper examines the equivalence between various set convergences, as studied in [7, 13, 22], induced by an arbitrary bornology S on a metric space $(X,d)$. Specifically, it focuses on the upper parts of the following set convergences: convergence deduced through uniform convergence of distance functionals on S (τS,d-convergence); convergence with respect to gap functionals determined by S (GS,d-convergence); and bornological convergence (S-convergence). In particular, we give necessary and sufficient conditions on the structure of the bornology S for the coincidence of τS,d⁺-convergence with GS,d⁺-convergence, as well as τS,d⁺-convergence with S⁺-convergence. A characterization for the equivalence of τS,d⁺-convergence and S⁺-convergence, in terms of certain convergence of nets, has also been given earlier by Beer, Naimpally, and Rodriguez-Lopez in [13]. To facilitate our study, we first devise new characterizations for τS,d⁺-convergence and S⁺-convergence, which we call their miss-type characterizations.