{
  "id": "1909.10953",
  "version": "v1",
  "published": "2019-09-24T14:24:07.000Z",
  "updated": "2019-09-24T14:24:07.000Z",
  "title": "Long colimits of topological groups II: Free groups and vector spaces",
  "authors": [
    "Rafael Dahmen",
    "Gábor Lukács"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological $\\mathbb{R}$-vector space $V(X)$ on $X$. We show that $V(X)$ is a quotient of the free abelian topological group on $[-1,1]\\times X$, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of $V(X)$ satisfy the so-called $\\textit{algebraic colimit property}$ defined in the authors' previous work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-24T14:24:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A16",
      "46M40",
      "46A99",
      "54D50",
      "54D55"
    ],
    "keywords": [
      "free groups",
      "long colimits",
      "free abelian topological group",
      "free topological group",
      "topological vector space analogues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}