{
  "id": "2207.08720",
  "version": "v1",
  "published": "2022-07-05T15:30:34.000Z",
  "updated": "2022-07-05T15:30:34.000Z",
  "title": "On Scott power spaces",
  "authors": [
    "Xiaoquan Xu",
    "Xinpeng Wen",
    "Xiaoyong Xi"
  ],
  "comment": "29 papes, 5 figures",
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper, we mainly discuss some basic properties of Scott power spaces. For a $T_0$ space $X$, let $\\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of $X$ endowed with the Smyth order. It is proved that the Scott power space $\\Sigma \\mathsf{K}(X)$ of a well-filtered space $X$ is still well-filtered, and a $T_0$ space $Y$ is well-filtered iff $\\Sigma \\mathsf{K}(Y)$ is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\\mathsf{K}(Y)$. A sober space is constructed for which its Scott power space is not sober. A few sufficient conditions are given under which a Scott power space is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces and Scott power spaces are also investigated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-05T15:30:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "54D99",
      "06B35",
      "06F30"
    ],
    "keywords": [
      "scott power space",
      "smyth power spaces",
      "upper vietoris topology",
      "nonempty compact saturated subsets",
      "scott topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}