{
  "id": "2204.05974",
  "version": "v1",
  "published": "2022-04-12T17:37:23.000Z",
  "updated": "2022-04-12T17:37:23.000Z",
  "title": "Baire property of some function spaces",
  "authors": [
    "Alexander V. Osipov",
    "Evgenii G. Pytkeev"
  ],
  "comment": "15 pages. arXiv admin note: text overlap with arXiv:2203.05976",
  "categories": [
    "math.GN"
  ],
  "abstract": "A compact space $X$ is called $\\pi$-monolithic if whenever a surjective continuous mapping $f:X\\rightarrow K$ where $K$ is a compacta there exists a compacta $T\\subseteq X$ such that $f(T)=K$. A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,Y)$ denote the space of all continuous $Y$- valued functions $C(X,Y)$ on a Tychonoff space $X$ with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space $X$ the space $C_p(X,\\{0,1\\})$ is Baire iff $C_p(X,K)$ is Baire whenever a $\\pi$-monolithic compact space $K$. For a Tychonoff space $X$ the space $C_p(X)$ is Baire iff $C_p(X,L)$ is Baire whenever a Frechet space $L$. We construct a totally disconnected Tychonoff space $T$ such that $C_p(T,M)$ is Baire for a separable metric space $M$ iff M is a Peano continuum. Moreover, the space $T$ such that $C_p(T,[0,1])$ is Baire but $Cp(T,\\{0,1\\})$ is not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-12T17:37:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "baire property",
      "function spaces",
      "open dense subsets",
      "monolithic compact space",
      "frechet space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}