{
  "id": "1809.06807",
  "version": "v1",
  "published": "2018-09-18T16:00:10.000Z",
  "updated": "2018-09-18T16:00:10.000Z",
  "title": "Disconnectedness properties of Hyperspaces",
  "authors": [
    "Rodrigo Hernández-Gutiérrez",
    "Angel Tamariz-Mascarúa"
  ],
  "journal": "Comment.Math.Univ.Carolin. 52,4 (2011) 569-591",
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $X$ be a Hausdorff space and let $\\mathcal{H}$ be one of the hyperspaces $CL(X)$, $\\mathcal{K}(X)$, $\\mathcal{F}(X)$ or $\\mathcal{F}_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\\mathcal{H}$: extremal disconnectedness, being a $F^\\prime$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\\subset X$ is a closed subset such that $(a)$ both $F$ and $X-F$ are totally disconnected, $(b)$ the quotient $X/F$ is hereditarily disconnected, then $\\mathcal{K}(X)$ is hereditarily disconnected. We also show an example proving that this result cannot be reversed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-18T16:00:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "54G05",
      "54G10",
      "54G12",
      "54G20"
    ],
    "keywords": [
      "disconnectedness properties",
      "hyperspaces",
      "main result states",
      "extremal disconnectedness",
      "vietoris topology"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}