Set Convergences via bornology
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914794148200448 |
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| author | Agarwal, Yogesh Jindal, Varun |
| author_facet | Agarwal, Yogesh Jindal, Varun |
| contents | This paper examines the equivalence between various set convergences, as studied in [7, 13, 22], induced by an arbitrary bornology $\mathcal{S}$ on a metric space $(X,d)$. Specifically, it focuses on the upper parts of the following set convergences: convergence deduced through uniform convergence of distance functionals on $\mathcal{S}$ ($τ_{\mathcal{S},d}$-convergence); convergence with respect to gap functionals determined by $\mathcal{S}$ ($G_{\mathcal{S},d}$-convergence); and bornological convergence ($\mathcal{S}$-convergence). In particular, we give necessary and sufficient conditions on the structure of the bornology $\mathcal{S}$ for the coincidence of $τ_{\mathcal{S},d}^+$-convergence with $\mathsf{G}_{\mathcal{S},d}^+$-convergence, as well as $τ_{\mathcal{S},d}^+$-convergence with $\mathcal{S}^+$-convergence. A characterization for the equivalence of $τ_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-convergence, in terms of certain convergence of nets, has also been given earlier by Beer, Naimpally, and Rodriguez-Lopez in [13]. To facilitate our study, we first devise new characterizations for $τ_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-convergence, which we call their miss-type characterizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07705 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Set Convergences via bornology Agarwal, Yogesh Jindal, Varun General Topology Functional Analysis 54B20, 54A10, 54E35, 54A20 This paper examines the equivalence between various set convergences, as studied in [7, 13, 22], induced by an arbitrary bornology $\mathcal{S}$ on a metric space $(X,d)$. Specifically, it focuses on the upper parts of the following set convergences: convergence deduced through uniform convergence of distance functionals on $\mathcal{S}$ ($τ_{\mathcal{S},d}$-convergence); convergence with respect to gap functionals determined by $\mathcal{S}$ ($G_{\mathcal{S},d}$-convergence); and bornological convergence ($\mathcal{S}$-convergence). In particular, we give necessary and sufficient conditions on the structure of the bornology $\mathcal{S}$ for the coincidence of $τ_{\mathcal{S},d}^+$-convergence with $\mathsf{G}_{\mathcal{S},d}^+$-convergence, as well as $τ_{\mathcal{S},d}^+$-convergence with $\mathcal{S}^+$-convergence. A characterization for the equivalence of $τ_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-convergence, in terms of certain convergence of nets, has also been given earlier by Beer, Naimpally, and Rodriguez-Lopez in [13]. To facilitate our study, we first devise new characterizations for $τ_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-convergence, which we call their miss-type characterizations. |
| title | Set Convergences via bornology |
| topic | General Topology Functional Analysis 54B20, 54A10, 54E35, 54A20 |
| url | https://arxiv.org/abs/2405.07705 |