{
  "abstract": "The Markov kernel based metric $D_1$ was introduced in 2011 in order to construct the scale-invariant dependence measure $ζ_1$, which assign each bivariate copula $C$ a dependence value in $[0,1]$, with $0$ exclusively for the case of independence, and $1$ exclusively for complete/functional dependence. In the original paper it has been shown that the resulting metric space $(\\mathcal{C},D_1)$ is separable and complete, however, no further topological properties were studied. Considering that $D_1$ has proved useful in a variety of contexts, using tools from infinite-dimensional topology, we here close this gap, show that $(\\mathcal{C},D_1)$ is homeomorphic to the Hilbert space $(\\ell_2,\\Vert \\cdot \\Vert_2)$, and prove that several subfamilies are either homeomorphic to $(\\ell_2,\\Vert \\cdot \\Vert_2)$ or to the Hilbert cube $(\\mathcal{H},ρ)$. Moreover, allowing for a better assessment of relative sizes, we show that various subfamilies are so-called $Z$-sets in $(\\mathcal{C},D_1)$, implying that they are topologically negligible in the full space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29275",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Fernández Sánchez"
    },
    {
      "name": "Wolfgang Trutschnig"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9knk7ph6amn8c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29275",
  "fulltext_url": null,
  "id": "9knk7ph6amn8c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29275",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Financial Risk and Volatility Modeling"
    ]
  ],
  "openalex": "W7214289230",
  "pdf_url": "https://arxiv.org/pdf/2609.29275",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9knk7ph6amn8c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29275",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29275"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On families of bivariate copulas and their interrelation with the Hilbert space l2 and the Hilbert cube H",
  "title_text": "On families of bivariate copulas and their interrelation with the Hilbert space l2 and the Hilbert cube H",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/9knk7ph6amn8c",
  "venue": "arXiv",
  "versions": []
}