{
  "id": "1901.04317",
  "version": "v1",
  "published": "2019-01-10T21:22:08.000Z",
  "updated": "2019-01-10T21:22:08.000Z",
  "title": "The wave model of metric spaces",
  "authors": [
    "M. I. Belishev",
    "S. A. Simonov"
  ],
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\Omega$ be a metric space, $A^t$ denote the metric neighborhood of the set $A\\subset\\Omega$ of the radius $t$; ${\\mathfrak O}$ be the lattice of open sets in $\\Omega$ with the partial order $\\subseteq$ and the order convergence. The lattice of $\\mathfrak O$-valued functions of $t\\in(0,\\infty)$ with the point-wise partial order and convergence contains the family ${I\\mathfrak O}=\\{A(\\cdot)\\,|\\,\\,A(t)=A^t,\\,\\,A\\in{\\mathfrak O}\\}$. Let $\\widetilde\\Omega$ be the set of atoms of the order closure $\\overline{I\\mathfrak O}$. We describe a class of spaces for which the set $\\widetilde\\Omega$, equipped with an appropriate metric, is isometric to the original space $\\Omega$. The space $\\widetilde\\Omega$ is the key element of the construction of the wave spectrum of a symmetric operator semi-bounded from below, which was introduced in a work of one of the authors. In that work, a program of constructing a functional model of operators of the aforementioned class was devised. The present paper is a step in realization of this program.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-10T21:22:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A55",
      "47A46",
      "06B35"
    ],
    "keywords": [
      "metric space",
      "wave model",
      "metric neighborhood",
      "functional model",
      "order convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}