{
  "id": "2002.08359",
  "version": "v1",
  "published": "2020-02-19T13:05:56.000Z",
  "updated": "2020-02-19T13:05:56.000Z",
  "title": "On decompositions of the real line",
  "authors": [
    "Gerald Kuba"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "Let X_t be a totally disconnected subset of the real line R for each t in R. We construct a partition {Y_t | t in R} of R into nowhere dense Lebesgue null sets Y_t such that for every t in R there exists an increasing homeomorphism from X_t onto Y_t. In particular, the real line can be partitioned into 2^{aleph_0} Cantor sets and also into 2^{aleph_0} mutually non-homeomorphic compact subspaces. Furthermore we prove that for every cardinal number k with 2 \\leq k \\leq 2^{aleph_0} the real line (as well as the Baire space R\\Q) can be partitioned into exactly k homeomorphic Bernstein sets and also into exactly k mutually non-homeomorphic Bernstein sets. We also investigate partitions of R into Marczewski sets, including the possibility that they are Luzin sets or Sierpinski sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-19T13:05:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A03",
      "54B05",
      "54B10"
    ],
    "keywords": [
      "real line",
      "decompositions",
      "dense lebesgue null sets",
      "mutually non-homeomorphic bernstein sets",
      "mutually non-homeomorphic compact subspaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}