On some convergence approach structures on hyperspaces
M. Ateş, F. Mynard, S. Sağıroğlu
TL;DR
The paper develops convergence-hyperspace structures for Cap spaces and their relation to convergence spaces, introducing upper and lower Kuratowski convergences and Fell-type topologies in the Cap framework. It shows that over an approach space, the lower Kuratowski convergence structure coincides with the $\vee$-Vietoris construction, while distinguishing a non-Archimedean upper-Fell structure that sits between the upper Kuratowski and earlier Fell-type structures. The Conv (coreflection) and Conv (reflection) correspondences tie $\lambda_{lK}$ to $c(\lambda)$ and $\lambda_{uK}$ to $r(\lambda)$, with topological reductions recovering classical Kuratowski convergence in the topological case. The work provides a Cap abstraction of results like: if the upper Kuratowski convergence over a topological space is pretopological, then it is topological, and it sets the stage for comparing when various hyperspace convergences coincide or differ in non-topological settings. Overall, this extends hyperspace convergence theory to the Cap setting, clarifying how coreflection/reflection influence Kuratowski and Fell constructions and offering a unified treatment across convergence- and approach-space frameworks.
Abstract
In the context of the category $\mathsf{Cap}$ 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 $\mathsf{Conv}$ of convergence spaces of a convergence approach space. In particular, over a pre-approach space, the $\mathsf{Conv}$-coreflection of the lower Kuratowski convergence approach structure is the lower Kuratowski convergence associated with the $\mathsf{Conv}$-coreflection of the base space, while the $\mathsf{Conv}$-reflection is the lower Kuratowski convergence associated with the $\mathsf{Conv}$-reflection. The $\mathsf{Conv}$-coreflection of the upper Kuratowski convergence approach is is the upper Kuratowski convergence associated with the $\mathsf{Conv}$-reflection of the base space, while the $\mathsf{Conv}$-reflection is the upper Kuratowski convergence associated with the $\mathsf{Conv}$-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 $\vee$-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 $\mathsf{Cap}$ abstraction of the classical result that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.