{
  "id": "2002.08026",
  "version": "v1",
  "published": "2020-02-19T06:42:18.000Z",
  "updated": "2020-02-19T06:42:18.000Z",
  "title": "On homeomorphism groups and the set-open topology",
  "authors": [
    "Alexander V. Osipov"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "In this paper we focus on the set-open topologies on the group $\\mathcal{H}(X)$ of all self-homeomorphisms of a topological space $X$ which yield continuity of both the group operations, product and inverse function. As a consequence, we make the more general case of Dijkstra's theorem. In this case a homogeneously encircling family $\\mathcal{B}$ consists of regular open sets and the closure of every set from $\\mathcal{B}$ is contained in the finite union of connected sets from $\\mathcal{B}$. Also we proved that the zero-cozero topology of $\\mathcal{H}(X)$ is the relativisation to $\\mathcal{H}(X)$ of the compact-open topology of $\\mathcal{H}(\\beta X)$ for any Tychonoff space $X$ and every homogeneous zero-dimensional space $X$ can be represented as the quotient space of a topological group with respect to a closed subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-19T06:42:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "54H11",
      "22A05"
    ],
    "keywords": [
      "set-open topology",
      "homeomorphism groups",
      "regular open sets",
      "dijkstras theorem",
      "yield continuity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}