{
  "id": "1310.4035",
  "version": "v3",
  "published": "2013-10-15T12:42:07.000Z",
  "updated": "2015-04-25T14:30:51.000Z",
  "title": "Chains of functions in $C(K)$-spaces",
  "authors": [
    "Tomasz Kania",
    "Richard J. Smith"
  ],
  "comment": "A revised version, 13 pp",
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "The Bishop property ($\\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pe{\\l}czy{\\'n}ski's classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after ($\\symbishop$) was first introduced. We show that if $\\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\\mathscr{D}$ has ($\\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindel\\\"of in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\\symbishop$) does not form a right ideal in $\\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-12T09:28:59.000Z",
      "abstract": "The Bishop property ($\\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pe\\lczy\\'nski's classic work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after {\\bish} was first introduced. We show that if $\\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\\mathscr{D}$ has ($\\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindel\\\"of in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\\symbishop$) does not form a right ideal in $\\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-25T14:30:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B50",
      "46E15",
      "37F20",
      "54F05",
      "46B26"
    ],
    "keywords": [
      "compact hausdorff spaces",
      "corson compact spaces",
      "gruenhage compact spaces",
      "results regarding local connectedness",
      "classic work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.4035K"
    }
  }
}