{
  "abstract": "Let $\\mathfrak{GH}$ denote the Gromov--Hausdorff class of isometry classes of nonempty metric spaces, equipped with the Gromov--Hausdorff extended pseudometric. We prove that every nonempty extended metric space is isometrically embeddable into $\\mathfrak{GH}$. More precisely, for every nonempty extended metric space $X$, all image points may be represented by metrics on a common set of cardinality $\\max\\{\\operatorname{dens} X,\\aleph_0\\}$. The main ingredient is a realization theorem for function spaces: for every nonempty set $I$, the space of all real-valued functions on $I$, endowed with the uniform extended metric, can be realized as a family of metrics on a fixed underlying set. Under this realization, the Gromov--Hausdorff distance agrees exactly with the uniform extended distance, including the case of infinite distance. The resulting metric spaces are complete, uniformly discrete, and unbounded, and have cardinality $\\max\\{|I|,\\aleph_0\\}$. If $I$ is countable, they may moreover be chosen proper and of bounded geometry. As further consequences, we obtain sharp cardinality results for clouds and universality results for cardinality-restricted parts of $\\mathfrak{GH}$ and for spheres in the associated zero-distance quotients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24279",
  "author_total": 1,
  "authors": [
    {
      "name": "Ryo Fukano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08etw571w5vnh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24279",
  "fulltext_url": null,
  "id": "08etw571w5vnh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24279",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223014806",
  "pdf_url": "https://arxiv.org/pdf/2610.24279",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08etw571w5vnh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.24279",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24279"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Every Extended Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Class",
  "title_text": "Every Extended Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Class",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/08etw571w5vnh",
  "venue": "arXiv",
  "versions": []
}