{
  "id": "1310.0678",
  "version": "v3",
  "published": "2013-10-02T12:24:49.000Z",
  "updated": "2014-05-30T10:04:40.000Z",
  "title": "The Stone-Cech compactifications of $ω^*\\setminus \\{x\\}$ and $S_κ\\setminus\\{x\\}$",
  "authors": [
    "Max F. Pitz",
    "Rolf Suabedissen"
  ],
  "comment": "18 pages. V3: Extended version; Thm 6.3 proves an open conjecture from V2",
  "categories": [
    "math.GN"
  ],
  "abstract": "The space $S_\\kappa$ is the Stone space of the $\\kappa$-saturated Boolean algebra of cardinality $\\kappa$. It exists provided that $\\kappa = \\kappa^{<\\kappa}$, and is characterised topologically as the unique $\\kappa$-Parovichenko space of weight $\\kappa$. Under the Continuum Hypothesis, $S_{\\omega_1}$ coincides with $\\omega^*$. This paper investigates questions related to the Stone-Cech compactification of spaces $S_\\kappa \\setminus \\{x\\}$, extending corresponding results obtained by Fine & Gillman and Comfort & Negrepontis for the space $\\omega^*$. We show that for every point $x$ of $S_\\kappa$, the Stone-Cech remainder of $S_\\kappa \\setminus \\{x\\}$ is a $\\kappa^+$-Parovichenko space of cardinality $2^{2^\\kappa}$ which admits a family of $2^\\kappa$ disjoint clopen sets. As a corollary we get that it is consistent with CH that the Stone-Cech remainders of $\\omega^* \\setminus \\{x\\}$ are all homeomorphic.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-30T10:04:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D40"
    ],
    "keywords": [
      "stone-cech compactification",
      "parovichenko space",
      "stone-cech remainder",
      "disjoint clopen sets",
      "stone space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.0678P"
    }
  }
}