{
  "id": "0710.1402",
  "version": "v3",
  "published": "2007-10-07T07:28:22.000Z",
  "updated": "2009-10-10T23:02:29.000Z",
  "title": "Covering an uncountable square by countably many continuous functions",
  "authors": [
    "Wiesław Kubiś",
    "Benjamin Vejnar"
  ],
  "comment": "Added new results (9 pages)",
  "journal": "Proc. Amer. Math. Soc. 140 (2012), no. 12, 4359--4368",
  "doi": "10.1090/S0002-9939-2012-11292-4",
  "categories": [
    "math.LO",
    "math.GN"
  ],
  "abstract": "We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\\times X$, where $X$ is an uncountable subset of the real line. This extends Sierpi\\'nski's theorem from 1919, saying that $S\\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if the size of $S$ does not exceed $\\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-10-10T23:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "03E15",
      "54H05"
    ],
    "keywords": [
      "uncountable square",
      "continuous functions",
      "planar borel sets",
      "extends sierpinskis theorem",
      "inverses cover"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.1402K"
    }
  }
}