{
  "id": "math/0407405",
  "version": "v1",
  "published": "2004-07-23T15:57:24.000Z",
  "updated": "2004-07-23T15:57:24.000Z",
  "title": "Half of an inseparable pair",
  "authors": [
    "Arnold W. Miller"
  ],
  "comment": "LaTex2e 16 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "A classical theorem of Luzin is that the separation principle holds for the Pi^0_alpha sets but fails for the Sigma^0_alpha sets. We show that for every Sigma^0_alpha set A which is not Pi^0_alpha there exists a Sigma^0_alpha set B which is disjoint from A but cannot be separated from A by a Delta^0_alpha set C. Assuming Pi^1_1-determancy it follows from a theorem of Steel that a similar result holds for Pi^1_1 sets. On the other hand assuming V=L there is a proper Pi^1_1 set which is not half of a Borel inseparable pair. These results answer questions raised by F.Dashiell. Latest version at: www.math.wisc.edu/~miller/",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-07-23T15:57:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "03E35",
      "03E60"
    ],
    "keywords": [
      "similar result holds",
      "separation principle holds",
      "borel inseparable pair",
      "results answer questions",
      "latest version"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7405M"
    }
  }
}