{
  "id": "1503.06703",
  "version": "v1",
  "published": "2015-03-23T16:12:45.000Z",
  "updated": "2015-03-23T16:12:45.000Z",
  "title": "The *-variation of Banach-Mazur game and forcing axioms",
  "authors": [
    "Yasuo Yoshinobu"
  ],
  "comment": "31 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We introduce a property of posets which strengthens the (\\omega_1+1)-strategic closedness. This property is defined using a variation of Banach-Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidor's theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from the (\\omega_1+1)-operational closedness, which we introduced in our previous work, by observing which portions of MA^+(\\omega_1-closed) are preserved or destroyed under forcing over posets with either property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-23T16:12:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E57",
      "03E35"
    ],
    "keywords": [
      "banach-mazur game",
      "forcing axioms",
      "first player chooses",
      "square principles",
      "closedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}