{
  "id": "1607.03576",
  "version": "v1",
  "published": "2016-07-13T02:45:45.000Z",
  "updated": "2016-07-13T02:45:45.000Z",
  "title": "Faithfulness of Directed Complete Posets based on Scott Closed Set Lattices",
  "authors": [
    "Dongsheng Zhao",
    "Luoshan Xu"
  ],
  "comment": "8 pages, Domains XII Workshop",
  "categories": [
    "math.GN"
  ],
  "abstract": "By Thron, a topological space $X$ has the property that $C(X)$ isomorphic to $C(Y)$ implies $X$ is homeomorphic to $Y$ iff $X$ is sober and $T_D$, where $C(X)$ and $C(Y)$ denote the lattices of closed sets of $X$ and $T_0$ space $Y$, respectively. When we consider dcpos (directed complete posets) equipped their Scott topologies, a similar question arises: which dcpos $P$ have the property that for any dcpo $Q$, $C_\\sigma(P)$ isomorphic to $C_\\sigma(Q)$ implies $P$ is isomorphic to $Q$ (such a dcpo $P$ will be called Scott closed set lattice faithful, or SCL-faithful in short)? Here $C_{\\sigma}(P)$ and $C_{\\sigma}(Q)$ denote the lattices of Scott closed sets of $P$ and $Q$, respectively. Following a characterization of continuous (quasicontinuous) dcpos in terms of $C_{\\sigma}(P)$, one easily deduces that every continuous (quasicontinuous) dcpo is SCL-faithful. Note that the Scott space of every continuous (quasicontinuous) dcpo is sober. Compared with Thron's result, one naturally asks whether every SCL-faithful dcpo is sober (with the Scott topology). In this paper we shall prove that some classes of dcpos are SCL-faithful, these classes contain some dcpos whose Scott topologies are not bounded sober. These results will help to obtain a complete characterization of SCL-faithful dcpos in the future.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-13T02:45:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06B30",
      "06B35",
      "54C35"
    ],
    "keywords": [
      "scott closed set lattice",
      "directed complete posets",
      "scott topology",
      "closed set lattice faithful",
      "faithfulness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}