{
  "id": "1606.04025",
  "version": "v1",
  "published": "2016-06-13T16:43:57.000Z",
  "updated": "2016-06-13T16:43:57.000Z",
  "title": "Topological properties of function spaces over ordinal spaces",
  "authors": [
    "Saak Gabriyelyan",
    "Jan Grebik",
    "Jerzy Kakol",
    "Lyubomyr Zdomskyy"
  ],
  "comment": "5 pages, comments are welcome",
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "Motivated by the classical Ascoli theorem, a topological space $X$ is said to be an Ascoli space if any compact subset $\\mathcal K$ of $C_k(Y)$ is evenly continuous. We study the $k_{\\mathbb R}$-property and the Ascoli property of $C_p(\\kappa)$ and $C_k(\\kappa)$ over ordinal spaces $\\kappa=[0,\\kappa)$. We prove that $C_p(\\kappa)$ is always an Ascoli space, while $C_p(\\kappa)$ is a $k_{\\mathbb R}$-space iff the cofinality of $\\kappa$ is countable. In particular, this provides the first $C_p$-example of an Ascoli space which is not a $k_{\\mathbb R}$-space, namely $C_p(\\omega_1)$. We show that $C_k(\\kappa)$ is Ascoli iff $\\mathrm{cf}(\\kappa)$ is countable iff $C_k(\\kappa)$ is metrizable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-13T16:43:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "54F05",
      "46A08",
      "54E18"
    ],
    "keywords": [
      "ordinal spaces",
      "function spaces",
      "topological properties",
      "ascoli space",
      "ascoli property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}