{
  "id": "1201.3814",
  "version": "v1",
  "published": "2012-01-18T15:08:38.000Z",
  "updated": "2012-01-18T15:08:38.000Z",
  "title": "The weights of closed subgroups of a locally compact group",
  "authors": [
    "Salvador Hernández",
    "Karl H. Hofmann",
    "Sidney A. Morris"
  ],
  "categories": [
    "math.GR",
    "math.GN"
  ],
  "abstract": "Let $G$ be an infinite locally compact group and $\\aleph$ a cardinal satisfying $\\aleph_0\\le\\aleph\\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\\aleph$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\\aleph$ a cardinal satisfying $\\aleph_0\\le\\aleph\\le \\lw(G)$, where $\\lw(G)$ is the local weight of $G$, there are either no infinite compact subgroups at all or there is a compact subgroup $N$ of $G$ with $w(N)=\\aleph$. (3) For an infinite abelian group $G$ there exists a properly ascending family of locally quasiconvex group topologies on $G$, say, $(\\tau_\\aleph)_{\\aleph_0\\le \\aleph\\le \\card(G)}$, such that $(G,\\tau_\\aleph)\\hat{\\phantom{m}}\\cong\\hat G$. Items (2) and (3) are shown in Section 5.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-18T15:08:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22C05",
      "22D05",
      "22D35"
    ],
    "keywords": [
      "closed subgroup",
      "infinite compact group contains",
      "infinite locally compact group",
      "infinite abelian group",
      "infinite closed metric subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.3814H"
    }
  }
}