{
  "id": "2407.16408",
  "version": "v1",
  "published": "2024-07-23T11:55:47.000Z",
  "updated": "2024-07-23T11:55:47.000Z",
  "title": "Set convergences and uniform convergence of distance functionals on a bornology",
  "authors": [
    "Yogesh Agarwal",
    "Varun Jindal"
  ],
  "comment": "25 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "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 $\\tau_{\\mathcal{S},d}$. 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 $(CL(X), \\tau_{\\mathcal{S},d})$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-23T11:55:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "54A10",
      "54D65",
      "54E35"
    ],
    "keywords": [
      "uniform convergence",
      "distance functionals",
      "set convergences",
      "unified approach",
      "hausdorff distance topologies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}