{
  "id": "0908.2228",
  "version": "v3",
  "published": "2009-08-16T10:05:59.000Z",
  "updated": "2009-11-20T07:57:32.000Z",
  "title": "The topological structure of direct limits in the category of uniform spaces",
  "authors": [
    "Taras Banakh",
    "Dusan Repovs"
  ],
  "comment": "10 pages",
  "journal": "Topology Appl. 157:6 (2010), 1091-1100",
  "doi": "10.1016/j.topol.2010.01.010",
  "categories": [
    "math.GN",
    "math.CT"
  ],
  "abstract": "Let $(X_n)_{n}$ be a sequence of uniform spaces such that each space $X_n$ is a closed subspace in $X_{n+1}$. We give an explicit description of the topology and uniformity of the direct limit $u-lim X_n$ of the sequence $(X_n)$ in the category of uniform spaces. This description implies that a function $f:u-lim X_n\\to Y$ to a uniform space $Y$ is continuous if for every $n$ the restriction $f|X_n$ is continuous and regular at the subset $X_{n-1}$ in the sense that for any entourages $U\\in\\U_Y$ and $V\\in\\U_X$ there is an entourage $V\\in\\U_X$ such that for each point $x\\in B(X_{n-1},V)$ there is a point $x'\\in X_{n-1}$ with $(x,x')\\in V$ and $(f(x),f(x'))\\in U$. Also we shall compare topologies of direct limits in various categories.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-11-20T07:57:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A13",
      "54B30",
      "54E15",
      "54H11"
    ],
    "keywords": [
      "uniform space",
      "direct limit",
      "topological structure",
      "explicit description",
      "compare topologies"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.2228B"
    }
  }
}