On some convergence approach structures on hyperspaces
M. Ate\c{s}, F. Mynard, S. Sa\u{g}{\i}ro\u{g}lu

TL;DR
This paper develops convergence structures on hyperspaces within convergence approach spaces, exploring their properties and relationships with classical topological concepts, and extends known results to this generalized setting.
Contribution
It introduces and analyzes approach analogs of Kuratowski and Fell convergences on hyperspaces in convergence approach spaces, linking them with coreflections and reflections.
Findings
Lower Kuratowski convergence coincides with the $igvee$-Vietoris structure over approach spaces.
Upper Fell convergence is coarser than upper Kuratowski but finer than upper Fell structures.
Classical topological results are extended to the convergence approach space setting.
Abstract
In the context of the category 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 of convergence spaces of a convergence approach space. In particular, over a pre-approach space, the -coreflection of the lower Kuratowski convergence approach structure is the lower Kuratowski convergence associated with the -coreflection of the base space, while the -reflection is the lower Kuratowski convergence associated with the -reflection. The -coreflection of the upper Kuratowski convergence approach is is the upper Kuratowski convergence associated with the -reflection of the base space, while the…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Banach Space Theory
