{
  "id": "1408.2137",
  "version": "v2",
  "published": "2014-08-09T17:04:52.000Z",
  "updated": "2015-04-27T12:49:15.000Z",
  "title": "Countable dense homogeneity in powers of zero-dimensional definable spaces",
  "authors": [
    "Andrea Medini"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "We show that, for a coanalytic subspace $X$ of $2^\\omega$, the countable dense homogeneity of $X^\\omega$ is equivalent to $X$ being Polish. This strengthens a result of Hru\\v{s}\\'ak and Zamora Avil\\'es. Then, inspired by results of Hern\\'andez-Guti\\'errez, Hru\\v{s}\\'ak and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of $2^\\omega$ such that $X^\\omega$ is countable dense homogeneous. This gives the first $\\mathsf{ZFC}$ answer to a question of Hru\\v{s}\\'ak and Zamora Avil\\'es. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space $X$ is included in a Polish subspace of $X$ then $X^\\omega$ is countable dense homogeneous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-09T17:04:52.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-27T12:49:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H05",
      "54E52",
      "54G20"
    ],
    "keywords": [
      "countable dense homogeneity",
      "zero-dimensional definable spaces",
      "zamora aviles",
      "countable dense homogeneous",
      "van mill"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2137M"
    }
  }
}