{
  "id": "2011.10261",
  "version": "v1",
  "published": "2020-11-20T08:14:40.000Z",
  "updated": "2020-11-20T08:14:40.000Z",
  "title": "Dominating and pinning down pairs for topological spaces",
  "authors": [
    "István Juhász",
    "Lajos Soukup",
    "Zoltán Szentmiklóssy"
  ],
  "comment": "16 pages",
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "We call a pair of infinite cardinals $(\\kappa,\\lambda)$ with $\\kappa > \\lambda$ a dominating (resp. pinning down) pair for a topological space $X$ if for every subset $A$ of $X$ (resp. family $\\mathcal{U}$ of non-empty open sets in $X$) of cardinality $\\le \\kappa$ there is $B \\subset X$ of cardinality $\\le \\lambda$ such that $A \\subset \\overline{B}$ (resp. $B \\cap U \\ne \\emptyset$ for each $U \\in \\mathcal{U}$). Clearly, a dominating pair is also a pinning down pair for $X$. Our definitions generalize the concepts introduced in [GTW] resp. [BT] which focused on pairs of the form $(2^\\lambda,\\lambda)$. The main aim of this paper is to answer a large number of the numerous problems from [GTW] and [BT] that asked if certain conditions on a space $X$ together with the assumption that $(2^\\lambda,\\lambda)$ or $((2^\\lambda)^+,\\lambda)$ is a pinning down pair or \\dominating pair for $X$ would imply $d(X) \\le \\lambda$. [BT] A. Bella, V.V. Tkachuk, Exponential density vs exponential domination, preprint [GTW] G. Gruenhage, V.V. Tkachuk, R.G. Wilson, Domination by small sets versus density, Topology and its Applications 282 (2020)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-20T08:14:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54A35",
      "54A65",
      "03E35"
    ],
    "keywords": [
      "topological space",
      "dominating",
      "non-empty open sets",
      "exponential domination",
      "exponential density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}