{
  "id": "math/9911122",
  "version": "v2",
  "published": "1999-11-17T02:00:57.000Z",
  "updated": "1999-11-30T19:12:43.000Z",
  "title": "A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "4 pages, LaTeX 209, a new theorem added (see Abstract and Note on p.2)",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let M denote the ideal of first category subsets of R. We prove that min{card X: X \\subseteq R, X \\not\\in M} is the smallest cardinality of a family S \\subseteq {0,1}^\\omega with the property that for each f: \\omega -> \\bigcup_{n \\in \\omega}{0,1}^n there exists a sequence {a_n}_{n \\in \\omega} belonging to S such that for infinitely many i \\in \\omega the infinite sequence {a_{i+n}}_{n \\in \\omega} extends the finite sequence f(i). We inform that S \\subseteq {0,1}^\\omega is not of first category if and only if for each f: \\omega -> \\bigcup_{n \\in \\omega}{0,1}^n there exists a sequence {a_n}_{n \\in \\omega} belonging to S such that for infinitely many i \\in \\omega the infinite sequence {a_{i+n}}_{n \\in \\omega} extends the finite sequence f(i).",
  "revisions": [
    {
      "version": "v2",
      "updated": "1999-11-30T19:12:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "54A25",
      "26A03"
    ],
    "keywords": [
      "combinatorial characterization",
      "minimal cardinality",
      "infinite sequence",
      "first category subsets",
      "smallest cardinality"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....11122T"
    }
  }
}