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 (ₒ, ₃-convergence) ; convergence with respect to gap functionals determined by S (Gₒ, ₃-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 ₒ, ₃^+-convergence with Gₒ, ₃^+-convergence, as well as ₒ, ₃^+-convergence with S^+-convergence. A characterization for the equivalence of ₒ, ₃^+-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 ₒ, ₃^+-convergence and S^+-convergence, which we call their miss-type characterizations.