{
  "id": "1701.03735",
  "version": "v1",
  "published": "2017-01-13T17:13:21.000Z",
  "updated": "2017-01-13T17:13:21.000Z",
  "title": "Classes of Polish spaces under effective Borel isomorphism",
  "authors": [
    "Vassilios Gregoriades"
  ],
  "journal": "Memoirs of the American Mathematical Society}, 240 (2016), no. 1135, vii+87",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study the equivalence classes under $\\Delta^1_1$ isomorphism, otherwise effective-Borel isomorphism, between complete separable metric spaces which admit a recursive presentation and we show the existence of strictly increasing and strictly decreasing sequences as well as of infinite antichains under the natural notion of $\\Delta^1_1$-reduction, as opposed to the non-effective case, where only two such classes exist, the one of the Baire space and the one of the naturals. A key tool for our study is a mapping $T \\mapsto \\mathcal{N}^T$ from the space of all trees on the naturals to the class of Polish spaces, for which every recursively presented space is $\\Delta^1_1$-isomorphic to some $\\mathcal{N}^T$ for a recursive $T$, so that the preceding spaces are representatives for the classes of $\\Delta^1_1$-isomorphism. We isolate two large categories of spaces of the type $\\mathcal{N}^T$, the Kleene spaces and the Spector-Gandy spaces and we study them extensively. Moreover we give results about hyperdegrees in the latter spaces and characterizations of the Baire space up to $\\Delta^1_1$-isomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-13T17:13:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "effective borel isomorphism",
      "polish spaces",
      "baire space",
      "complete separable metric spaces",
      "large categories"
    ],
    "tags": [
      "journal article",
      "monograph"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Mem. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}