{
  "id": "2004.05321",
  "version": "v1",
  "published": "2020-04-11T06:35:04.000Z",
  "updated": "2020-04-11T06:35:04.000Z",
  "title": "Topological properties of some function spaces",
  "authors": [
    "Saak Gabriyelyan",
    "Alexander V. Osipov"
  ],
  "comment": "39 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\\mathcal{I}}$-Tychonoff space for some ideal $\\mathcal{I}$ of compact sets of $X$. Denote by $C_{\\mathcal{I}}(X,Y)$ the space of continuous functions from $X$ to $Y$ endowed with the $\\mathcal{I}$-open topology. We prove that $C_{\\mathcal{I}}(X,Y)$ is Fr\\'{e}chet - Urysohn iff $X$ has the property $\\gamma_{\\mathcal{I}}$. We characterize zero - dimensional Tychonoff spaces $X$ for which the space $C_{\\mathcal{I}}(X,{\\bf 2})$ is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if $Y$ is not compact, then $C_{p}(X,Y)$ is Fr\\'{e}chet - Urysohn iff it is sequential iff it is a $k$-space iff $X$ has the property $\\gamma$. An analogous result is obtained for the space of bounded continuous functions taking values in a metrizable locally convex space. Denote by $B_{1}(X,Y)$ and $B(X,Y)$ the space of Baire one functions and the space of all Baire functions from $X$ to $Y$, respectively. If $H$ is a subspace of $B(X,Y)$ containing $B_{1}(X,Y)$, then $H$ is metrizable iff it is a $\\sigma$ - space iff it has countable $cs^*$ - character iff $X$ is countable. If additionally $Y$ is not compact, then $H$ is Fr\\'{e}chet - Urysohn iff it is sequential iff it is a $k$ - space iff it has countable tightness iff $X_{\\aleph_0}$ has the property $\\gamma$, where $X_{\\aleph_0}$ is the space $X$ with the Baire topology. We show that if $X$ is a Polish space, then the space $B_{1}(X,\\mathbb{R})$ is normal iff $X$ is countable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-11T06:35:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "function spaces",
      "topological properties",
      "continuous functions",
      "sequential",
      "dimensional tychonoff spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}