{
  "id": "1709.07998",
  "version": "v1",
  "published": "2017-09-23T04:07:30.000Z",
  "updated": "2017-09-23T04:07:30.000Z",
  "title": "On the weak tightness, Hausdorff spaces, and power homogeneous compacta",
  "authors": [
    "Nathan Carlson"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Motivated by results of Juh\\'asz and van Mill in [13], we define the cardinal invariant $wt(X)$, the weak tightness of a topological space $X$, and show that $|X|\\leq 2^{L(X)wt(X)\\psi(X)}$ for any Hausdorff space $X$ (Theorem 2.8). As $wt(X)\\leq t(X)$ for any space $X$, this generalizes the well-known cardinal inequality $|X|\\leq 2^{L(X)t(X)\\psi(X)}$ for Hausdorff spaces (Arhangel{\\cprime}ski\\u{i}~[1],\\v{S}}apirovski\\u{i}}~[18]) in a new direction. Theorem 2.8 is generalized further using covers by $G_\\kappa$-sets, where $\\kappa$ is a cardinal, to show that if $X$ is a power homogeneous compactum with a countable cover of dense, countably tight subspaces then $|X|\\leq\\mathfrak{c}$, the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-23T04:07:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54B10"
    ],
    "keywords": [
      "power homogeneous compactum",
      "hausdorff space",
      "weak tightness",
      "well-known cardinal inequality",
      "countably tight subspaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}