{
  "id": "1610.09245",
  "version": "v1",
  "published": "2016-10-28T14:46:34.000Z",
  "updated": "2016-10-28T14:46:34.000Z",
  "title": "On the cardinality of Hausdorff spaces and H-closed spaces",
  "authors": [
    "Nathan Carlson",
    "Jack Porter"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "We introduce the cardinal invariant $aL^\\prime(X)$ and show that $|X|\\leq 2^{aL^\\prime(X)\\chi(X)}$ for any Hausdorff space $X$ (a corollary of Theorem 4.4. This invariant has the properties a) $aL^\\prime(X)=\\aleph_0$ if $X$ is H-closed, and b) $aL(X)\\leq aL^\\prime(X)\\leq aL_c(X)$. Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound $2^{L(X)\\chi(X)}$ from which it follows that $|X|\\leq 2^{\\psi_c(X)}$ if $X$ is H-closed (Dow/Porter [5]). The invariant $aL^\\prime(X)$ is constructed using convergent open ultrafilters and an operator $c:\\scr{P}(X)\\to\\scr{P}(X)$ with the property $clA\\subseteq c(A)\\subseteq cl_\\theta(A)$ for all $A\\subseteq X$. As a comparison with this open ultrafilter approach, in $\\S 3$ we additionally give a $\\kappa$-filter variation of Hodel's proof [10] of the Dow-Porter result. Finally, for an infinite cardinal $\\kappa$, in $\\S 5$ we introduce $\\kappa$wH-closed spaces, $\\kappa H^\\prime$-closed spaces, and $\\kappa H^{\\prime\\prime}$-closed spaces. The first two notions generalize the H-closed property. Key results in this connection are that a) if $\\kappa$ is an infinite cardinal and $X$ a $\\kappa$wH-closed space with a dense set of isolated points such that $\\chi(X)\\leq\\kappa$, then $|X|\\leq 2^{\\kappa}$, and b) if $X$ is $\\kappa H^\\prime$-closed or $\\kappa H^{\\prime\\prime}$-closed then $aL^\\prime(X)\\leq\\kappa$. This latter result relates these notions to the invariant $aL^\\prime(X)$ and the operator $c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-28T14:46:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D20",
      "54A25",
      "54D10"
    ],
    "keywords": [
      "hausdorff space",
      "h-closed spaces",
      "cardinality",
      "infinite cardinal",
      "convergent open ultrafilters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}