{
  "id": "2011.14299",
  "version": "v1",
  "published": "2020-11-29T06:20:02.000Z",
  "updated": "2020-11-29T06:20:02.000Z",
  "title": "A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications",
  "authors": [
    "Jerzy Kakol",
    "Arkady Leiderman"
  ],
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $\\Delta$-space in the sense of \\cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a \\v{C}ech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mr\\'owka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,\\omega_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $\\Delta$-spaces is invariant under basic topological operations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-29T06:20:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "54G12",
      "54H05",
      "46A03"
    ],
    "keywords": [
      "application",
      "locally convex space",
      "basic topological operations",
      "separable compact space",
      "tychonoff space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}