{
  "id": "1504.01785",
  "version": "v1",
  "published": "2015-04-08T00:03:43.000Z",
  "updated": "2015-04-08T00:03:43.000Z",
  "title": "Cardinalities of weakly Lindelöf spaces with regular $G_κ$-diagonals",
  "authors": [
    "Ivan S. Gotchev"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "For a Urysohn space $X$ we define the \\emph{regular diagonal degree} $\\overline{\\Delta}(X)$ of $X$ to be the minimal infinite cardinal $\\kappa$ such that $X$ has a regular $G_\\kappa$-diagonal i.e. there is a family $(U_\\eta:\\eta<\\kappa)$ of open sets in $X^2$ such that $\\{(x,x)\\in X^2:x\\in X\\} = \\cap_{\\eta<\\kappa} \\overline{U}_\\eta$. In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\\leq 2^{wL(X)\\cdot\\overline{\\Delta}(X)}$; (2) $|X|\\le wL(X)^{\\overline{\\Delta}(X)\\cdot\\chi(X)}$; and (3) $|X|\\le aL(X)^{\\overline{\\Delta}(X)}$. It follows from (1) that every weakly Lindel\\\"of space with a regular $G_\\delta$-diagonal has cardinality at most $2^\\omega$. This generalizes Buzyakova's result that the cardinality of a ccc-space with a regular $G_\\delta$-diagonal does not exceed $2^\\omega$. It also follows that $|X|\\leq 2^{\\chi(X)\\cdot wL(X)\\cdot\\overline{\\Delta}(X)}$; hence Bell, Ginsburg and Woods inequality $|X|\\le 2^{\\chi(X)wL(X)}$, which is known to be true for normal $T_1$-spaces, is true also for a large class of Urysohn spaces that includes all spaces with regular $G_\\delta$-diagonals. Inequality (2) implies that when $X$ is a space with a regular $G_\\delta$-diagonal then $|X|\\le wL(X)^{\\chi(X)}$. This improves significantly Bell, Ginsburg and Woods inequality for the class of normal spaces with regular $G_\\delta$-diagonals. In particular (2) shows that the cardinality of every first countable space with a regular $G_\\delta$-diagonal does not exceed $wL(X)^\\omega$. For the class of spaces with regular $G_\\delta$-diagonals (3) improves Bella and Cammaroto inequality $|X|\\le 2^{\\chi(X)\\cdot aL(X)}$, which is valid for all Urysohn spaces. Also, it follows from (3) that the cardinality of every space with a regular $G_\\delta$-diagonal does not exceed $aL(X)^\\omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-08T00:03:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54D20"
    ],
    "keywords": [
      "weakly lindelöf spaces",
      "urysohn space",
      "cardinality",
      "woods inequality",
      "generalizes buzyakovas result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}