On some convergence approach structures on hyperspaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917270848012288 |
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| author | Ateş, M. Mynard, F. Sağıroğlu, S. |
| author_facet | Ateş, M. Mynard, F. Sağıroğlu, S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12149 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On some convergence approach structures on hyperspaces Ateş, M. Mynard, F. Sağıroğlu, S. General Topology 54A20, 54B20, 54A05, 54H99, 18F99 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. |
| title | On some convergence approach structures on hyperspaces |
| topic | General Topology 54A20, 54B20, 54A05, 54H99, 18F99 |
| url | https://arxiv.org/abs/2602.12149 |