{
  "id": "1506.04665",
  "version": "v1",
  "published": "2015-06-15T16:51:22.000Z",
  "updated": "2015-06-15T16:51:22.000Z",
  "title": "Regular $G_δ$-diagonals and some upper bounds for cardinality of topological spaces",
  "authors": [
    "Ivan S. Gotchev",
    "Mikhail G. Tkachenko",
    "Vladimir V. Tkachuk"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "We recall that for a Urysohn space $X$ the \\emph{regular diagonal degree $\\overline{\\Delta}(X)$ of $X$} is defined as the minimal infinite cardinal $\\kappa$ such that $X$ has a regular $G_\\kappa$-diagonal. The \\emph{o-tightness} of $X$ does not exceed $\\kappa$, or $\\mathrm{ot}(X)\\le\\kappa$, if for every family $\\mathcal{V}$ of open subsets of $X$ and for every point $x\\in X$ with $x\\in\\overline{\\bigcup \\mathcal{V}}$ there exists a subfamily $\\mathcal{V}\\subset \\mathcal{U}$ such that $|\\mathcal{V}|\\le\\kappa$ and $x\\in\\overline{\\bigcup\\mathcal{V}}$. We will say that the \\emph{dense o-tightness} of $X$ does not exceed $\\kappa$, or $\\mathrm{dot}(X)\\le\\kappa$, if for every family $\\mathcal{U}$ of open subsets of $X$ whose union is dense in $X$ and for every point $x\\in X$ there exists a subfamily $\\mathcal{V}\\subset\\mathcal{U}$ such that $|\\mathcal{V}|\\le\\kappa$ and $x\\in\\overline{\\bigcup\\mathcal{V}}$. The main results of this paper are as follows: If $X$ is a Urysohn space then (1) $|X|\\le wL(X)^{\\pi\\chi(X)\\cdot\\overline{\\Delta}(X)}$ and (2) $|X|\\le wL(X)^{s\\Delta_2(X)\\cdot{\\mathrm{dot}(X)}}$; (3) if $2^\\omega=\\omega_1$ and $X$ is a space with a regular $G_\\delta$-diagonal and caliber $\\omega_1$ then $X$ is separable; if $X$ is a Hausdorff space then (4) $|X|\\le\\pi w(X)^{\\mathrm{ot}(X)\\cdot\\psi_c(X)}$ and (5) $|X|\\le \\pi\\chi(X)^{\\mathrm{ot}(X)\\cdot\\psi_c(X)\\cdot aL_c(X)}$. Inequalities (1) and (2) generalize some resent results obtained by I. S. Gotchev and D. Basile, A. Bella, and G. J. Ridderbos. As a corollary of (3) we obtain (under CH) a generalization and a positive answer respectively of a result and a question of R. Buzyakova. We use (4) to prove (5) which gives another generalization of the famous Arhangel'skii's inequality that the cardinality of any Hausdorff space $X$ does not exceed $2^{\\chi(X)\\cdot L(X)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-15T16:51:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54D10",
      "54D20"
    ],
    "keywords": [
      "upper bounds",
      "topological spaces",
      "cardinality",
      "urysohn space",
      "open subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150604665G"
    }
  }
}