{
  "id": "2003.06542",
  "version": "v1",
  "published": "2020-03-14T03:29:58.000Z",
  "updated": "2020-03-14T03:29:58.000Z",
  "title": "Coincidence of the upper Vietoris topology and the Scott topology",
  "authors": [
    "Xiaoquan Xu",
    "Zhongqiang Yang"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "For a $T_0$ space $X$, let $\\mk (X)$ be the poset of all compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if for any $K_1, K_2\\in \\mk (X)$, $K_2\\ll K_1$ in $\\mk (X)$ if{}f $K_2\\subseteq \\ii~\\!K_1$. In this paper, we give several connections among the well-filteredness of $X$, the sobriety of $X$, the local compactness of $X$, the core compactness of $X$, the property Q of $X$, the coincidence of the upper Vietoris topology and Scott topology on $\\mk (X)$, and the continuity of $x\\mapsto\\ua x : X \\longrightarrow \\Sigma~\\!\\! \\mk (X)$ (where $\\Sigma~\\!\\! \\mk (X)$ is the Scott space of $\\mk (X)$). It is shown that for a well-filtered space $X$ for which its Smyth power space $P_S(X)$ is first-countable, the following three properties are equivalent: the local compactness of $X$, the core compactness of $X$ and the continuity of $\\mk (X)$. It is also proved that for a first-countable $T_0$ space $X$ in which the set of minimal elements of $K$ is countable for any compact saturated subset $K$ of $X$, the Smyth power space $P_S(X)$ is first-countable. For the Alexandroff double circle $Y$, which is Hausdorff and first-countable, we show that its Smyth power space $P_S(Y)$ is not first-countable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-14T03:29:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper vietoris topology",
      "scott topology",
      "smyth power space",
      "coincidence",
      "core compactness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}