{
  "id": "2003.06293",
  "version": "v1",
  "published": "2020-03-13T13:46:56.000Z",
  "updated": "2020-03-13T13:46:56.000Z",
  "title": "All $\\mathbb Q$-gluons are homeomorphic",
  "authors": [
    "Taras Banakh",
    "Yaryna Stelmakh"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper we discover a (super)connected countable Hausdorf space called a $\\mathbb Q$-gluon and prove its topogical uniqueness. A topological spaces $X$ is called $\\mathbb Q$-$gluon$ if it possesses a decreasing sequence $(X_n)_{n\\in\\omega}$ of nonempty closed subsets such that $X_0=X$, $\\bigcap_{n\\in\\omega}X_n=\\emptyset$, for every $n\\in\\omega$ the complement $X\\setminus X_{n+1}$ is homeomorphic to $\\mathbb Q$, and for any non-empty open set $U\\subseteq X_n$ the closure $\\overline U$ contains some set $X_m$. Our main results says that any two $\\mathbb Q$-gluons are homeomorphic. We shall find examples of $\\mathbb Q$-gluons among quotient spaces, orbit spaces of group actions, and projective spaces of some topological vector spaces over countable topological fields. In particular, we prove that the projective space $\\mathbb Q\\mathsf P^\\infty$ of the topological vector space $\\mathbb Q^{<\\omega}=\\{(x_n)_{n\\in\\omega}\\in\\mathbb Q^\\omega:|\\{n\\in\\omega:x_n\\ne0\\}|<\\omega\\}$ is a $\\mathbb Q$-gluon.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-13T13:46:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54F65",
      "54G15",
      "51H10"
    ],
    "keywords": [
      "homeomorphic",
      "topological vector space",
      "projective space",
      "non-empty open set",
      "main results says"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}