{
  "id": "1706.04911",
  "version": "v1",
  "published": "2017-06-15T15:01:50.000Z",
  "updated": "2017-06-15T15:01:50.000Z",
  "title": "Finite powers of selectively pseudocompact groups",
  "authors": [
    "S. Garcia-Ferreira",
    "A. H. Tomita"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "A space $X$ is called {\\it selectively pseudocompact} if for each sequence $(U_{n})_{n\\in \\mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{n\\in \\mathbb{N}}$ of points in $X$ such that $cl_X(\\{x_n : n < \\omega\\}) \\setminus \\big(\\bigcup_{n < \\omega}U_n \\big) \\neq \\emptyset$ and $x_{n}\\in U_{n}$, for each $n < \\omega$. Countably compact space spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of $CH$, that for every positive integer $k > 2$ there exists a topological group whose $k$-th power is countably compact but its $(k+1)$-st power is not selectively pseudocompact. This provides a positive answer to a question posed in \\cite{gt} in any model of $ZFC+CH$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-15T15:01:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H11",
      "54B05",
      "54E99"
    ],
    "keywords": [
      "selectively pseudocompact groups",
      "finite powers",
      "pairwise disjoint nonempty open subsets",
      "countably compact space spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}