Set Convergences via bornology

Abstract

This paper examines the equivalence between various set convergences, as studied in [7, 13, 22], induced by an arbitrary bornology $\mathcal{S}$ 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 $\mathcal{S}$ ($\tau_{\mathcal{S},d}$-convergence); convergence with respect to gap functionals determined by $\mathcal{S}$ ($G_{\mathcal{S},d}$-convergence); and bornological convergence ($\mathcal{S}$-convergence). In particular, we give necessary and sufficient conditions on the structure of the bornology $\mathcal{S}$ for the coincidence of $\tau_{\mathcal{S},d}^+$-convergence with $\mathsf{G}_{\mathcal{S},d}^+$-convergence, as well as $\tau_{\mathcal{S},d}^+$-convergence with $\mathcal{S}^+$-convergence. A characterization for the equivalence of $\tau_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-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 $\tau_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-convergence, which we call their miss-type characterizations.