{
  "id": "1107.5383",
  "version": "v2",
  "published": "2011-07-27T05:05:29.000Z",
  "updated": "2012-07-05T02:02:06.000Z",
  "title": "Borel's Conjecture in Topological Groups",
  "authors": [
    "Fred Galvin",
    "Marion Scheepers"
  ],
  "comment": "15 pages",
  "categories": [
    "math.LO",
    "math.GN",
    "math.GR"
  ],
  "abstract": "We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\\kappa$, let {\\sf BC}$_{\\kappa}$ denote this generalization. Then ${\\sf BC}_{\\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\\neg{\\sf BC}_{\\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\\aleph_1$. Using the connection of ${\\sf BC}_{\\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\\sf BC}_{\\aleph_1}$. (2)If it is consistent that ${\\sf BC}_{\\aleph_1}$ holds, then it is consistent that there is an inaccessible cardinal. (3)If it is consistent that there is a 1-inaccessible cardinal with $\\omega$ inaccessible cardinals above it, then $\\neg{\\sf BC}_{\\aleph_{\\omega}} \\, +\\, (\\forall n<\\omega){\\sf BC}_{\\aleph_n}$ is consistent. (4)If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\\sf BC}_{\\aleph_{\\omega}}$. (5)If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\\sf BC}_{\\kappa}$ holds for a proper class of cardinals $\\kappa$ of countable cofinality.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-05T02:02:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "03E35",
      "03E55",
      "03E65",
      "22A99"
    ],
    "keywords": [
      "borels conjecture",
      "consistent",
      "topological groups",
      "classical borel conjecture",
      "infinite cardinal number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.5383G"
    }
  }
}