{
  "id": "1705.04564",
  "version": "v1",
  "published": "2017-05-07T14:02:40.000Z",
  "updated": "2017-05-07T14:02:40.000Z",
  "title": "First Order Theories of Some Lattices of Open Sets",
  "authors": [
    "Oleg Kudinov",
    "Victor Selivanov"
  ],
  "comment": "18 pages",
  "categories": [
    "math.LO",
    "cs.LO"
  ],
  "abstract": "We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\\mathbb{R}^n$, $n\\geq1$, and the domain $P\\omega$) this theory is $m$-equivalent to first order arithmetic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-07T14:02:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D78",
      "03D45",
      "03D55",
      "03D30"
    ],
    "keywords": [
      "first order theory",
      "first order arithmetic",
      "second order arithmetic",
      "natural computable metric spaces",
      "natural topological spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}