Set convergences and uniform convergence of distance functionals on a bornology
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| Format: | Preprint |
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2024
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| _version_ | 1866916333438894080 |
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| author | Agarwal, Yogesh Jindal, Varun |
| author_facet | Agarwal, Yogesh Jindal, Varun |
| contents | For a metric space $(X,d)$, Beer, Naimpally, and Rodriguez-Lopez in ([17]) proposed a unified approach to explore set convergences via uniform convergence of distance functionals on members of an arbitrary family $\mathcal{S}$ of subsets of $X$. The associated topology on the collection $CL(X)$ of all nonempty closed subsets of $(X,d)$ is denoted by $τ_{\mathcal{S},d}$. As special cases, this unified approach includes classical Wijsman, Attouch-Wets, and Hausdorff distance topologies. In this article, we investigate various topological characteristics of the hyperspace $(CL(X), τ_{\mathcal{S},d})$ when $\mathcal{S}$ is a bornology on $(X,d)$. In order to do this, a new class of bornologies and a new metric topology on $CL(X)$ have been introduced and studied. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16408 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Set convergences and uniform convergence of distance functionals on a bornology Agarwal, Yogesh Jindal, Varun General Topology 54B20, 54A10, 54D65, 54E35 For a metric space $(X,d)$, Beer, Naimpally, and Rodriguez-Lopez in ([17]) proposed a unified approach to explore set convergences via uniform convergence of distance functionals on members of an arbitrary family $\mathcal{S}$ of subsets of $X$. The associated topology on the collection $CL(X)$ of all nonempty closed subsets of $(X,d)$ is denoted by $τ_{\mathcal{S},d}$. As special cases, this unified approach includes classical Wijsman, Attouch-Wets, and Hausdorff distance topologies. In this article, we investigate various topological characteristics of the hyperspace $(CL(X), τ_{\mathcal{S},d})$ when $\mathcal{S}$ is a bornology on $(X,d)$. In order to do this, a new class of bornologies and a new metric topology on $CL(X)$ have been introduced and studied. |
| title | Set convergences and uniform convergence of distance functionals on a bornology |
| topic | General Topology 54B20, 54A10, 54D65, 54E35 |
| url | https://arxiv.org/abs/2407.16408 |