Kantorovich-Rubinstein quasi-metrics IV: Lenses, quasi-lenses and forks

Lenses and quasi-lenses on a space X form models of erratic non-determinism. When X is equipped with a quasi-metric d, there are natural quasi-metrics dP and dPa on the space of quasi-lenses on X, which resemble the Pompeiu-Hausdorff metric (and contain it as a subcase when d is a metric), and are tightly connected to the Kantorovich-Rubinstein quasi-metrics dKR and dKRa of Parts I, II and III, through an isomorphism between quasi-lenses and so-called discrete normalized forks. We show that the space of quasi-lenses on X is continuous complete, resp. algebraic complete, if X,d is itself continuous complete, resp. algebraic complete. In those cases, we also show that the dP-Scott and dPa-Scott topologies coincide with the Vietoris topology. We then prove similar results on spaces of (sub)normalized forks, not necessarily discrete; those are models of mixed erratic non-determinism and probabilistic choice. For that, we need the additional assumption that the cone LX of lower continuous maps from X to R‾+, with the Scott topology, has an almost open addition map (which is the case if X is locally compact and coherent, notably); we also need X to be compact in the case of normalized forks. The relevant quasi-metrics are simple extensions of the Kantorovich-Rubinstein quasi-metrics dKRa of Parts I, II and III.