{
  "id": "1704.07740",
  "version": "v1",
  "published": "2017-04-25T15:34:07.000Z",
  "updated": "2017-04-25T15:34:07.000Z",
  "title": "Selectively pseudocompact groups without non-trivial convergent sequences",
  "authors": [
    "Dmitri Shakhmatov",
    "Víctor Hugo Yañez"
  ],
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent sequences having the following \"selective\" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first author. As a by-product, we show that the free precompact Boolean group over any disjoint sum of maximal countable spaces contains no infinite compact subsets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-25T15:34:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22A05",
      "54A20",
      "54D30",
      "54H11"
    ],
    "keywords": [
      "non-trivial convergent sequences",
      "selectively pseudocompact groups",
      "free precompact boolean group",
      "major open problem",
      "maximal countable spaces contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}