{
  "id": "math/0310438",
  "version": "v1",
  "published": "2003-10-28T14:51:46.000Z",
  "updated": "2003-10-28T14:51:46.000Z",
  "title": "Ultrafilters with property (s)",
  "authors": [
    "Arnold W. Miller"
  ],
  "comment": "LaTeX2e 10 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "A set X which is a subset of the Cantor set has property (s) (Marczewski (Spzilrajn)) iff for every perfect set P there exists a perfect set Q contained in P such that Q is a subset of X or Q is disjoint from X. Suppose U is a nonprincipal ultrafilter on omega. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s) in the ground model. It is known that selective ultrafilters or even P-points are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that assuming CH (or more generally MA for ctble posets) there exists an ultrafilter U with property (s) such that U does not generate an ultrafilter in any extension which adds a new subset of omega. http://www.math.wisc.edu/~miller/res/index.html miller@math.wisc.edu",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-10-28T14:51:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E17",
      "03E50"
    ],
    "keywords": [
      "perfect set",
      "partial order",
      "generic extension",
      "cantor set",
      "sacks forcing"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....10438M"
    }
  }
}