{
  "id": "1311.6544",
  "version": "v1",
  "published": "2013-11-26T02:46:59.000Z",
  "updated": "2013-11-26T02:46:59.000Z",
  "title": "The non-Urysohn number of a topological space",
  "authors": [
    "Ivan S. Gotchev"
  ],
  "comment": "8 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "We call a nonempty subset $A$ of a topological space $X$ finitely non-Urysohn if for every nonempty finite subset $F$ of $A$ and every family $\\{U_x:x\\in F\\}$ of open neighborhoods $U_x$ of $x\\in F$, $\\cap\\{\\mathrm{cl}(U_x):x\\in F\\}\\ne\\emptyset$ and we define the non-Urysohn number of $X$ as follows: $nu(X):=1+\\sup\\{|A|:A$ is a finitely non-Urysohn subset of $X\\}$. Then for any topological space $X$ and any subset $A$ of $X$ we prove the following inequalities: (1) $|\\mathrm{cl}_\\theta(A)|\\le |A|^{\\kappa(X)}\\cdot nu(X)$, (2) $|[A]_\\theta|\\le (|A|\\cdot nu(X))^{\\kappa(X)}$, (3) $|X|\\le nu(X)^{\\kappa(X)sL_\\theta(X)}$, and (4) $|X|\\le nu(X)^{\\kappa(X)aL(X)}$. In 1979, A. V. Arhangelskii asked if the inequality $|X|\\le 2^{\\chi(X)wL_c(X)}$ was true for every Hausdorff space $X$. It follows from the third inequality that the answer of this question is in the affirmative for all spaces with $nu(X)$ not greater than the cardinality of the continuum. We also give a simple example of a Hausdorff space $X$ such that $|\\mathrm{cl}_\\theta(A)|>|A|^{\\chi(X)}U(X)$ and $|\\mathrm{cl}_\\theta(A)|>(|A|\\cdot U(X))^{\\chi(X)}$, where $U(X)$ is the Urysohn number of $X$, recently introduced by Bonanzinga, Cammaroto and Matveev. This example shows that in (1) and (2) above, $nu(X)$ cannot be replaced by $U(X)$ and answers some questions posed by Bella and Cammaroto (1988), Bonanzinga, Cammaroto and Matveev (2011), and Bonanzinga and Pansera (2012).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-26T02:46:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54D10"
    ],
    "keywords": [
      "topological space",
      "non-urysohn number",
      "hausdorff space",
      "inequality",
      "nonempty finite subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.6544G"
    }
  }
}