{
  "id": "0901.0175",
  "version": "v1",
  "published": "2009-01-01T11:24:45.000Z",
  "updated": "2009-01-01T11:24:45.000Z",
  "title": "Convergent sequences in minimal groups",
  "authors": [
    "Dmitri Shakhmatov"
  ],
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "A Hausdorff topological group G is minimal if every continuous isomorphism f : G --> H between G and a Hausdorff topological group H is open. Clearly, every compact Hausdorff group is minimal. It is well known that every infinite compact Hausdorff group contains a non-trivial convergent sequence. We extend this result to minimal abelian groups by proving that every infinite minimal abelian group contains a non-trivial convergent sequence. Furthermore, we show that \"abelian\" is essential and cannot be dropped. Indeed, for every uncountable regular cardinal kappa we construct a Hausdorff group topology T_kappa on the free group F(kappa) with kappa many generators having the following properties: (i) (F(kappa), T_kappa) is a minimal group; (ii) every subset of F(kappa) of size less than kappa is T_kappa-discrete (and thus also T_kappa-closed); (iii) there are no non-trivial proper T_kappa-closed normal subgroups of F(kappa). In particular, all compact subsets of (F(kappa), T_kappa) are finite, and every Hausdorff quotient group of (F(kappa), T_kappa) is minimal (that is, (F(kappa), T_kappa) is totally minimal).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-01-01T11:24:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22A05",
      "22C05",
      "54A10",
      "54A20",
      "54A25",
      "54D25",
      "54H11"
    ],
    "keywords": [
      "minimal group",
      "non-trivial convergent sequence",
      "infinite compact hausdorff group contains",
      "hausdorff topological group",
      "infinite minimal abelian group contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0901.0175S"
    }
  }
}