{
  "id": "1610.07842",
  "version": "v1",
  "published": "2016-10-25T12:18:44.000Z",
  "updated": "2016-10-25T12:18:44.000Z",
  "title": "Recovering a compact Hausdorff space $X$ from the compatibility ordering on $C(X)$",
  "authors": [
    "Tomasz Kania",
    "Martin Rmoutil"
  ],
  "comment": "17 pp",
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "Let $f$ and $g$ be scalar-valued, continuous functions on some topological space. We say that $g$ dominates $f$ in the compatibility ordering if $g$ coincides with $f$ on the support of $f$. We prove that two compact Hausdorff spaces are homeomorphic if and only if there exists a compatibility isomorphism between their families of scalar-valued, continuous functions. We derive the classical theorems of Gelfand-Kolmogorov, Milgram and Kaplansky as easy corollaries to our result as well as a theorem of Jarosz (Bull. Canad. Math. Soc. 1990) thereby building~a common roof for these theorems. Sharp automatic-continuity results for compatibility isomorphisms are also established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-25T12:18:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E10",
      "46T20",
      "06F25"
    ],
    "keywords": [
      "compact hausdorff space",
      "compatibility ordering",
      "compatibility isomorphism",
      "continuous functions",
      "sharp automatic-continuity results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}