{
  "id": "2211.10765",
  "version": "v1",
  "published": "2022-11-19T18:33:06.000Z",
  "updated": "2022-11-19T18:33:06.000Z",
  "title": "On Baire property, compactness and completeness properties of spaces of Baire functions",
  "authors": [
    "Alexander V. Osipov"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let $\\alpha$ be a countable ordinal. Characterize topological spaces $X$ and $Y$ for which the function space $B_{\\alpha}(X,Y)$ is Baire. In this paper, for any Frechet space $Y$ , we have obtained a characterization topological spaces $X$ for which the function space $B_{\\alpha}(X,Y)$ is Baire. In particular, we proved that $B_{\\alpha}(X,\\mathbb{R})$ is Baire if and only if $B_{\\alpha}(X,Y)$ is Baire for any Banach space $Y$. Also we proved that many completeness and compactness properties coincide in spaces $B_{\\alpha}(X,Y)$ for any Frechet space $Y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-19T18:33:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "baire functions",
      "baire property",
      "completeness properties",
      "function space",
      "frechet space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}