{
  "id": "2405.07705",
  "version": "v1",
  "published": "2024-05-13T12:47:39.000Z",
  "updated": "2024-05-13T12:47:39.000Z",
  "title": "Set Convergences via bornology",
  "authors": [
    "Yogesh Agarwal",
    "Varun Jindal"
  ],
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "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 $\\mathcal{S}$ ($\\tau_{\\mathcal{S},d}$-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 $\\tau_{\\mathcal{S},d}^+$-convergence with $\\mathsf{G}_{\\mathcal{S},d}^+$-convergence, as well as $\\tau_{\\mathcal{S},d}^+$-convergence with $\\mathcal{S}^+$-convergence. A characterization for the equivalence of $\\tau_{\\mathcal{S},d}^+$-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 $\\tau_{\\mathcal{S},d}^+$-convergence and $\\mathcal{S}^+$-convergence, which we call their miss-type characterizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-13T12:47:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "54A10",
      "54E35",
      "54A20"
    ],
    "keywords": [
      "set convergences",
      "miss-type characterizations",
      "equivalence",
      "sufficient conditions",
      "arbitrary bornology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}