{
  "id": "1703.10144",
  "version": "v1",
  "published": "2017-03-29T17:19:38.000Z",
  "updated": "2017-03-29T17:19:38.000Z",
  "title": "Continuous reducibility and dimension of metric spaces",
  "authors": [
    "Philipp Schlicht"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space $(X,d)$ of positive dimension, there are uncountably many Borel subsets of $(X,d)$ that are pairwise incomparable with respect to continuous reducibility. The reducibility that is given by the collection of continuous functions on a topological space $(X,\\tau)$ is called the \\emph{Wadge quasi-order} for $(X,\\tau)$. We further show that this quasi-order, restricted to the Borel subsets of a Polish space $(X,\\tau)$, is a \\emph{well-quasiorder (wqo)} if and only if $(X,\\tau)$ has dimension $0$, as an application of the main result. Moreover, we give further examples of applications of the technique, which is based on a construction of graph colorings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-29T17:19:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous reducibility",
      "borel subsets",
      "main result",
      "quasi-order",
      "polish metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}