{
  "id": "1709.04234",
  "version": "v1",
  "published": "2017-09-13T10:17:07.000Z",
  "updated": "2017-09-13T10:17:07.000Z",
  "title": "Borel subsets of the real line and continuous reducibility",
  "authors": [
    "Daisuke Ikegami",
    "Philipp Schlicht",
    "Hisao Tanaka"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We study classes of Borel subsets of the real line $\\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on $\\mathbb{R}$. Notably, we explore several structural properties of Borel subsets of $\\mathbb{R}$ that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order $\\omega_1$, its reverse $\\omega_1^\\star$ and the poset $\\mathcal{P}(\\omega)/\\mathsf{fin}$ of inclusion modulo finite error. As a consequence of its proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to $\\mathbb{Q}$. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In our second main theorem, we introduce a combinatorial property that is shown to characterize those $F_\\sigma$ sets that are reducible to $\\mathbb{Q}$. This is applied to construct a minimal set below $\\mathbb{Q}$ and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-13T10:17:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel subsets",
      "real line",
      "continuous reducibility",
      "wadge quasi-order",
      "inclusion modulo finite error"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}