{
  "id": "1708.03455",
  "version": "v1",
  "published": "2017-08-11T07:16:50.000Z",
  "updated": "2017-08-11T07:16:50.000Z",
  "title": "A new construction of universal spaces for asymptotic dimension",
  "authors": [
    "G. C. Bell",
    "A. Nagórko"
  ],
  "journal": "Topology and its Applications Volume 160, Issue 1, Pages 159-169 (2013)",
  "doi": "10.1016/j.topol.2012.10.016",
  "categories": [
    "math.GT"
  ],
  "abstract": "For each $n$, we construct a separable metric space $\\mathbb{U}_n$ that is universal in the coarse category of separable metric spaces with asymptotic dimension ($\\mathop{asdim}$) at most $n$ and universal in the uniform category of separable metric spaces with uniform dimension ($\\mathop{udim}$) at most $n$. Thus, $\\mathbb{U}_n$ serves as a universal space for dimension $n$ in both the large-scale and infinitesimal topology. More precisely, we prove: \\[ \\mathop{asdim} \\mathbb{U}_n = \\mathop{udim} \\mathbb{U}_n = n \\] and such that for each separable metric space $X$, a) if $\\mathop{asdim} X \\leq n$, then $X$ is coarsely equivalent to a subset of $\\mathbb{U}_n$; b) if $\\mathop{udim} X \\leq n$, then $X$ is uniformly homeomorphic to a subset of $\\mathbb{U}_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-11T07:16:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54F45",
      "54E15"
    ],
    "keywords": [
      "separable metric space",
      "asymptotic dimension",
      "universal space",
      "construction",
      "infinitesimal topology"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}