{
  "id": "1805.12454",
  "version": "v1",
  "published": "2018-05-31T13:16:31.000Z",
  "updated": "2018-05-31T13:16:31.000Z",
  "title": "The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications",
  "authors": [
    "Carmelo A. Finocchiaro",
    "Marco Fontana",
    "Dario Spirito"
  ],
  "comment": "to apper in the Rocky Mountain J. Math",
  "categories": [
    "math.GN",
    "math.AC"
  ],
  "abstract": "Given an arbitrary spectral space $X$, we consider the set ${\\boldsymbol{\\mathcal{X}}}(X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on ${\\boldsymbol{\\mathcal{X}}}(X)$ and, after observing that it coincides the upper Vietoris topology, we prove that ${\\boldsymbol{\\mathcal{X}}}(X)$ is itself a spectral space, that this construction is functorial, and that ${\\boldsymbol{\\mathcal{X}}}(X)$ provides an extension of $X$ in a more `complete' spectral space. Among the applications, we show that, starting from an integral domain $D$, ${\\boldsymbol{\\mathcal{X}}}(\\mathrm{Spec}(D))$ is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on $D$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-31T13:16:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13A10",
      "13A15",
      "13B10",
      "13G05",
      "14A05",
      "54A10",
      "54F65"
    ],
    "keywords": [
      "upper vietoris topology",
      "inverse-closed subsets",
      "applications",
      "arbitrary spectral space",
      "nonempty subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}