Set Convergences via Bornology

This paper examines the equivalence of various set convergences that are induced by an arbitrary bornology on a metric space. Specifically, it focuses on the upper parts of the following set convergences: convergence deduced through uniform convergence of distance functionals on bornology; convergence with respect to gap functionals determined by bornology; and bornological convergence. Our primary attention here is on the first of these set convergences, namely the upper part of convergence that comes via distance functionals. As main results, we give necessary and sufficient conditions on the structure of the bornology for this set convergence to coincide with upper gap and upper bornological convergences. To facilitate our study, we first devise new characterizations for these convergences (other than upper gap convergence), which we call their miss-type characterizations.