{
  "id": "1909.03357",
  "version": "v1",
  "published": "2019-09-08T00:48:51.000Z",
  "updated": "2019-09-08T00:48:51.000Z",
  "title": "A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences",
  "authors": [
    "Artur Hideyuki Tomita"
  ],
  "journal": "Topology Appl. Volume 259, 1 June 2019, Pages 347-364",
  "doi": "10.1016/j.topol.2019.02.040",
  "categories": [
    "math.GN"
  ],
  "abstract": "We show that if $\\kappa \\leq \\omega$ and there exists a group topology without non-trivial convergent sequences on an Abelian group $H$ such that $H^n$ is countably compact for each $n<\\kappa$ then there exists a topological group $G$ such that $G^n$ is countably compact for each $n <\\kappa$ and $G^{\\kappa}$ is not countably compact. If in addition $H$ is torsion, then the result above holds for $\\kappa=\\omega_1$. Combining with other results in the literature, we show that: $a)$ Assuming ${\\mathfrak c}$ incomparable selective ultrafilters, for each $n \\in \\omega$, there exists a group topology on the free Abelian group $G$ such that $G^n$ is countably compact and $G^{n+1}$ is not countably compact. (It was already know for $\\omega$). $b)$ If $\\kappa \\in \\omega \\cup \\{\\omega\\} \\cup \\{\\omega_1\\}$, there exists in ZFC a topological group $G$ such that $G^\\gamma$ is countably compact for each cardinal $\\gamma <\\kappa$ and $G^\\kappa$ is not countably compact.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-08T00:48:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "van douwen-like zfc theorem",
      "non-trivial convergent sequences",
      "countably compact groups",
      "small powers",
      "abelian group"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}