{
  "id": "2304.03455",
  "version": "v1",
  "published": "2023-04-07T03:03:11.000Z",
  "updated": "2023-04-07T03:03:11.000Z",
  "title": "The quasi-metrizability of hyperspaces",
  "authors": [
    "Chuan Liu",
    "Fucai Lin"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "For a space $X$, let $(CL(X), \\tau_V)$, $(CL(X), \\tau_{locfin})$ and $(CL(X), \\tau_F)$ be the set $CL(X)$ of all nonempty closed subsets of $X$ which are endowed with Vietoris topology, locally finite topology and Fell topology respectively. We prove that $(CL(X), \\tau_V)$ is quasi-metrizable if and only if $X$ is a separable metrizable space and the set of all non-isolated points of $X$ is compact, $(CL(X), \\tau_{locfin})$ is quasi-metrizable or symmetrizable if and only if $X$ is metrizable and the set of all non-isolated points of $X$ is compact, and $(CL(X), \\tau_F)$ is quasi-metrizable if and only if $X$ is hemicompact and metrizable. As an application, we give a negative answer to a Conjecture in \\cite{LL2022}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-07T03:03:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "54E35",
      "54E45"
    ],
    "keywords": [
      "quasi-metrizability",
      "hyperspaces",
      "non-isolated points",
      "nonempty closed subsets",
      "fell topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}