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