{
  "id": "1005.1149",
  "version": "v1",
  "published": "2010-05-07T08:06:18.000Z",
  "updated": "2010-05-07T08:06:18.000Z",
  "title": "The Markov-Zariski topology of an abelian group",
  "authors": [
    "Dikran Dikranjan",
    "Dmitri Shakhmatov"
  ],
  "journal": "Journal of Algebra, 324 (2010), 1125-1158",
  "doi": "10.1016/j.jalgebra.2010.04.025",
  "categories": [
    "math.GR",
    "math.AG",
    "math.GN"
  ],
  "abstract": "According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (=totally bounded) Hausdorff group topology on G. The family of all algebraic subsets of an abelian group G forms the family of closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or verbal, topology of G. We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Frechet-Urysohn. For a countable family F of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology T on G such that the T-closure of each member of F coincides with its Zariski closure. As an application, we provide a characterization of the subsets of G that are dense in some Hausdorff group topology on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-07T08:06:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20A99",
      "20A45",
      "22A05",
      "54H11"
    ],
    "keywords": [
      "abelian group",
      "markov-zariski topology",
      "elementary algebraic set",
      "hausdorff group topology",
      "precompact metric group topology"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1149D"
    }
  }
}