{
  "id": "1412.2949",
  "version": "v1",
  "published": "2014-12-09T13:12:18.000Z",
  "updated": "2014-12-09T13:12:18.000Z",
  "title": "On the Borel Complexity of Characterized Subgroups",
  "authors": [
    "Dikran Dikranjan",
    "Daniele Impieri"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In a compact abelian group $X$, a characterized subgroup is a subgroup $H$ such that there exists a sequence of characters $\\vs=(v_n)$ of $X$ such that $H=\\{x\\in X:v_n(x)\\to 0 \\text{ in } \\T\\}$. Gabriyelyan proved for $X=\\T$, that $\\{x\\in\\T:n!x\\to 0 \\text{ in }\\T\\}$ is not an $F_\\sigma$-set. In this paper, we give a complete description of the $F_\\sigma$-subgroups of $\\T$ characterized by sequences of integers $\\vs=(v_n)$ such that $v_n|v_{n+1}$ for all $n\\in\\N$ (we show that these are exactly the countable characterized subgroups). Moreover in the general setting of compact metrizable abelian groups, we give a new point of view to study the Borel complexity of characterized subgroups in terms of appropriate test-topologies in the whole group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-09T13:12:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22C05"
    ],
    "keywords": [
      "borel complexity",
      "compact abelian group",
      "compact metrizable abelian groups",
      "complete description",
      "appropriate test-topologies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.2949D"
    }
  }
}