{
  "id": "2009.13273",
  "version": "v1",
  "published": "2020-09-28T13:03:14.000Z",
  "updated": "2020-09-28T13:03:14.000Z",
  "title": "Metric Segments in Gromov--Hausdorff class",
  "authors": [
    "Olga Borisova"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "We study properties of metric segments in the class of all metric spaces considered up to an isometry, endowed with Gromov--Hausdorff distance. On the isometry classes of all compact metric spaces, the Gromov-Hausdorff distance is a metric. A metric segment is a class that consists of points lying between two given ones. By von Neumann--Bernays--G\\\"odel (NBG) axiomatic set theory, a proper class is a \"monster collection\", e.g., the collection of all cardinal sets. We prove that any metric segment in the proper class of isometry classes of all metric spaces with the Gromov-Hausdorff distance is a proper class if the segment contains at least one metric space at positive distances from the segment endpoints. In addition, we show that the restriction of a non-degenerated metric segment to compact metric spaces is a non-compact set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T13:03:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gromov-hausdorff class",
      "gromov-hausdorff distance",
      "compact metric spaces",
      "proper class",
      "isometry classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}