{
  "id": "2104.10506",
  "version": "v1",
  "published": "2021-04-21T12:52:48.000Z",
  "updated": "2021-04-21T12:52:48.000Z",
  "title": "Basic properties of $X$ for which spaces $C_p(X)$ are distinguished",
  "authors": [
    "Jerzy Kakol",
    "Arkady Leiderman"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In our paper [18] we showed that a Tychonoff space $X$ is a $\\Delta$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $\\Delta$ of $\\Delta$-spaces is invariant under the basic topological operations. We prove that if $X \\in \\Delta$ and $\\varphi:X \\to Y$ is a continuous surjection such that $\\varphi(F)$ is an $F_{\\sigma}$-set in $Y$ for every closed set $F \\subset X$, then also $Y\\in \\Delta$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_i\\in \\Delta$, then also $X\\in \\Delta$. In particular, $\\sigma$-product of any family of scattered Eberlein compact spaces is a $\\Delta$-space and the product of a $\\Delta$-space with a countable space is a $\\Delta$-space. Our results give answers to several open problems posed in \\cite{KL}. Let $T:C_p(X) \\longrightarrow C_p(Y)$ be a continuous linear surjection. We observe that $T$ admits an extension to a linear continuous operator $\\widehat{T}$ from $R^X$ onto $R^Y$ and deduce that $Y$ is a $\\Delta$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is. Making use of obtained results, we provide a very short proof for the claim that every compact $\\Delta$-space has countable tightness. As a consequence, under Proper Forcing Axiom (PFA) every compact $\\Delta$-space is sequential. In the article we pose a dozen open questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-21T12:52:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "basic properties",
      "dozen open questions",
      "scattered eberlein compact spaces",
      "locally convex space",
      "short proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}