{
  "id": "2202.08615",
  "version": "v1",
  "published": "2022-02-17T11:57:46.000Z",
  "updated": "2022-02-17T11:57:46.000Z",
  "title": "The topological type of spaces consisting of certain metrics on locally compact metrizable spaces with the compact-open topology",
  "authors": [
    "Katsuhisa Koshino"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "For a separable locally compact but not compact metrizable space $X$, let $\\alpha X = X \\cup \\{x_\\infty\\}$ be the one-point compactification with the point at infinity $x_\\infty$. We denote by $EM(X)$ the space consisting of admissible metrics on $X$, which can be extended to an admissible metric on $\\alpha X$, endowed with the compact-open topology. Let $ \\mathbf{c}_0 \\subset (0,1)^\\mathbb{N}$ be the space of sequences converging to $0$. In this paper, we shall show that if $X$ is separable, locally connected and locally compact but not compact, and there exists a sequence $\\{C_i\\}$ of connected sets in $X$ such that for all positive integers $i, j \\in \\mathbb{N}$ with $|i - j| \\leq 1$, $C_i \\cap C_j \\neq \\emptyset$, and for each compact set $K \\subset X$, there is a positive integer $i(K) \\in \\mathbb{N}$ such that for any $i \\geq i(K)$, $C_i \\subset X \\setminus K$, then $EM(X)$ is homeomorphic to $ \\mathbf{c}_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-17T11:57:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "57N20",
      "54E35",
      "54E40",
      "54E45"
    ],
    "keywords": [
      "locally compact metrizable spaces",
      "compact-open topology",
      "topological type",
      "spaces consisting",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}