{
  "id": "1406.7805",
  "version": "v1",
  "published": "2014-06-30T16:37:24.000Z",
  "updated": "2014-06-30T16:37:24.000Z",
  "title": "Between countably compact and $ω$-bounded",
  "authors": [
    "István Juhász",
    "Lajos Soukup",
    "Zoltán Szentmiklóssy"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \\in \\{\\omega D, \\omega N, C_2 \\}$ where $\\omega D \\, \\equiv$ \"countable discrete\", $\\omega N \\, \\equiv$ \"countable nowhere dense\", and $C_2 \\,\\equiv$ \"second countable\". Clearly, for each of these $P$-bounded is between countably compact and $\\omega$-bounded. We give examples in ZFC that separate all these boundedness properties and their appropriate combinations. Consistent separating examples with better properties (such as: smaller cardinality or weight, local compactness, first countability) are also produced. We have interesting results concerning $\\omega D$-bounded spaces which show that $\\omega D$-boundedness is much stronger than countable compactness: $\\bullet$ Regular $\\omega D$-bounded spaces of Lindel\\\"of degree $< cov(\\mathcal{M})$ are $\\omega$-bounded. $\\bullet$ Regular $\\omega D$-bounded spaces of countable tightness are $\\omega N$-bounded, and if $\\mathfrak{b} > \\omega_1$ then even $\\omega$-bounded. $\\bullet$ If a product of Hausdorff space is $\\omega D$-bounded then all but one of its factors must be $\\omega$-bounded. $\\bullet$ Any product of at most $\\mathfrak{t}$ many Hausdorff $\\omega D$-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-30T16:37:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54A35",
      "54D30",
      "54D65"
    ],
    "keywords": [
      "bounded spaces",
      "hausdorff space",
      "compact closure",
      "first countability",
      "local compactness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.7805J"
    }
  }
}