{
  "id": "1504.04198",
  "version": "v1",
  "published": "2015-04-16T12:06:26.000Z",
  "updated": "2015-04-16T12:06:26.000Z",
  "title": "Topological properties of function spaces $C_k(X,2)$ over zero-dimensional metric spaces $X$",
  "authors": [
    "S. Gabriyelyan"
  ],
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "Let $X$ be a zero-dimensional metric space and $X'$ its derived set. We prove the following assertions: (1) the space $C_k(X,2)$ is an Ascoli space iff $C_k(X,2)$ is $k_\\mathbb{R}$-space iff either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (2) $C_k(X,2)$ is a $k$-space iff either $X$ is a topological sum of a Polish locally compact space and a discrete space or $X$ is not locally compact but $X'$ is compact, (3) $C_k(X,2)$ is a sequential space iff $X$ is a Polish space and either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (4) $C_k(X,2)$ is a Fr\\'{e}chet--Urysohn space iff $C_k(X,2)$ is a Polish space iff $X$ is a Polish locally compact space, (5) the space $C_k(X,2)$ is normal iff $X'$ is separable. In cases (1)-(3) we obtain also a topological and algebraical structure of $C_k(X,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-16T12:06:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "54D50",
      "54D55"
    ],
    "keywords": [
      "zero-dimensional metric space",
      "function spaces",
      "topological properties",
      "polish locally compact space",
      "polish space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150404198G"
    }
  }
}