{
  "id": "1501.02877",
  "version": "v1",
  "published": "2015-01-13T03:44:48.000Z",
  "updated": "2015-01-13T03:44:48.000Z",
  "title": "Density character of subgroups of topological groups",
  "authors": [
    "Arkady Leiderman",
    "Sidney A. Morris",
    "Mikhail G. Tkachenko"
  ],
  "comment": "21 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, locally compact abelian groups and connected locally compact groups and is closed under products and closed subgroups. A topological group G is almost connected if the quotient group of G by the connected component of its identity is compact. We prove that an almost connected pro-Lie group is separable iff its weight is not greater than c. It is deduced that an almost connected pro-Lie group is separable if and only if it is a subspace of a separable Hausdorff space. It is proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. Every precompact topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact group of weight c. This implies that there is a wealth of closed nonseparable subgroups of separable pseudocompact groups. An example is presented under CH of a separable countably compact abelian group which contains a non-separable closed subgroup. It is proved that the following conditions are equivalent for an omega-narrow topological group G: (i) G is a subspace of a separable regular space; (ii) G is a subgroup of a separable topological group; (iii) G is a closed subgroup of a separable pathconnected locally pathconnected group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-13T03:44:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D65",
      "22D05"
    ],
    "keywords": [
      "topological group",
      "density character",
      "closed subgroup",
      "countably compact abelian group",
      "locally compact"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150102877L"
    }
  }
}