  For a metric space $(X,d)$, Beer, Naimpally, and Rodriguez-Lopez in ([17])
proposed a unified approach to explore set convergences via uniform convergence
of distance functionals on members of an arbitrary family $\mathcal{S}$ of
subsets of $X$. The associated topology on the collection $CL(X)$ of all
nonempty closed subsets of $(X,d)$ is denoted by $\tau_{\mathcal{S},d}$. As
special cases, this unified approach includes classical Wijsman, Attouch-Wets,
and Hausdorff distance topologies. In this article, we investigate various
topological characteristics of the hyperspace $(CL(X), \tau_{\mathcal{S},d})$
when $\mathcal{S}$ is a bornology on $(X,d)$. In order to do this, a new class
of bornologies and a new metric topology on $CL(X)$ have been introduced and
studied.
