{
  "id": "1102.4328",
  "version": "v2",
  "published": "2011-02-21T20:58:23.000Z",
  "updated": "2013-11-04T08:22:27.000Z",
  "title": "The groups $S^3$ and $SO(3)$ have no invariant binary $k$-network",
  "authors": [
    "Taras Banakh",
    "Slawomir Turek"
  ],
  "comment": "5 pages",
  "journal": "Topology Proc. 41 (2013) 261-266",
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "A family $\\mathcal N$ of closed subsets of a topological space $X$ is called a {\\em closed $k$-network} if for each open set $U\\subset X$ and a compact subset $K\\subset U$ there is a finite subfamily $\\mathcal F\\subset\\mathcal N$ with $K\\subset\\bigcup\\F\\subset \\mathcal N$. A compact space $X$ is called {\\em supercompact} if it admits a closed $k$-network $\\mathcal N$ which is {\\em binary} in the sense that each linked subfamily $\\mathcal L\\subset\\mathcal N$ is centered. A closed $k$-network $\\mathcal N$ in a topological group $G$ is {\\em invariant} if $xAy\\in\\mathcal N$ for each $A\\in\\mathcal N$ and $x,y\\in G$. According to a result of Kubi\\'s and Turek, each compact (abelian) topological group admits an (invariant) binary closed $k$-network. In this paper we prove that the compact topological groups $S^3$ and $\\SO(3)$ admit no invariant binary closed $k$-network.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-04T08:22:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D30",
      "22C05"
    ],
    "keywords": [
      "invariant binary",
      "open set",
      "compact space",
      "compact subset"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.4328B"
    }
  }
}