In the context of the category Cap of convergence approach spaces and contractions, we introduce and study approach analogs of the upper and lower Kuratowski convergences, upper-Fell and Fell topologies on the set of closed subsets of the coreflection on the category Conv of convergence spaces of a convergence approach space. In particular, over a pre-approach space, the Conv-coreflection of the lower Kuratowski convergence approach structure is the lower Kuratowski convergence associated with the Conv-coreflection of the base space, while the Conv-reflection is the lower Kuratowski convergence associated with the Conv-reflection. The Conv-coreflection of the upper Kuratowski convergence approach is is the upper Kuratowski convergence associated with the Conv-reflection of the base space, while the Conv-reflection is the upper Kuratowski convergence associated with the Conv-coreflection of the base space. We show that, over an approach space, the lower Kuratowski convergence approach structure is in fact an approach structure that coincides with the ∨ -Vietoris approach structure introduced by Lowen and his collaborators, though it may be strictly finer over a general convergence approach space. We show that the upper Fell convergence approach structure is a non-Archimedean approach structure coarser than the upper Kuratowski convergence approach, but finer than the upper Fell approach structure introduced by the first and third author. We also obtain a Cap abstraction of the classical result that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.
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