Bornological convergences and local proximity spaces

In this paper we clarify the intensive interaction among uniformity, proximity and bornology in local proximity spaces bringing up their underlying uniform characters. By using uniformity and bornology, we give a procedure as an exhaustive method to generate local proximity spaces. We show that the hyperspace of all non-empty closed subsets of a local proximity space carries a very appropriate Fell-type topology, which admits a formulation as hit and far-miss topology and also characterizes as the topology of a two-sided uniform bornological convergence. Furthermore, that topology is induced by the weak uniformity generated by infimum value functionals of the real functions which preserve proximity and boundedness and have a bounded support. Finally, we give necessary and sufficient conditions for topologies of two-sided uniform bornological convergences agree. In particular, we focus on the class of uniform spaces carrying a linearly ordered base, known also as ωμ-metric spaces. Equipping the hyperspace of an ωμ-metric space with the Attouch–Wets or bounded Hausdorff topology in the usual way, we achieve among others in the ωμ-metric setting the following two issues. The former: the Attouch–Wets topologies associated with two ωμ-metrics on a same space coincide if and only if they have the same bounded sets and are uniformly equivalent on any bounded set. The latter: the Attouch–Wets topologies associated with two ωμ-metrics on a same space coincide if and only if their hit and bounded far-miss topologies agree.