{
  "id": "2002.00409",
  "version": "v1",
  "published": "2020-02-02T14:52:53.000Z",
  "updated": "2020-02-02T14:52:53.000Z",
  "title": "Constructing a coarse space with a given Higson or binary corona",
  "authors": [
    "Taras Banakh",
    "Igor Protasov"
  ],
  "comment": "20 pages",
  "categories": [
    "math.GN",
    "math.MG"
  ],
  "abstract": "For any compact Hausdorff space $K$ we construct a canonical finitary coarse structure $\\mathcal E_{X,K}$ on the set $X$ of isolated points of $K$. This construction has two properties: $\\bullet$ If a finitary coarse space $(X,\\mathcal E)$ is metrizable, then its coarse structure $\\mathcal E$ coincides with the coarse structure $\\mathcal E_{X,\\bar X}$ generated by the Higson compactification $\\bar X$ of $X$; $\\bullet$ A compact Hausdorff space $K$ coincides with the Higson compactification of the coarse space $(X,\\mathcal E_{X,K})$ if the set $X$ is dense in $K$ and the space $K$ is Frechet-Urysohn. This implies that a compact Hausdorff space $K$ is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) $K$ is perfectly normal; (ii) $K$ has weight $w(K)\\le\\omega_1$ and character $\\chi(K)<\\mathfrak p$. Under CH every (zero-dimensional) compact Hausdorff space of weight $\\le\\omega_1$ is homeomorphic to the Higson (resp. binary) corona of some cellular finitary coarse space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-02T14:52:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D30",
      "54D35",
      "54D40",
      "54F05",
      "54E15",
      "54E35"
    ],
    "keywords": [
      "compact hausdorff space",
      "binary corona",
      "cellular finitary coarse space",
      "higson compactification",
      "canonical finitary coarse structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}