{
  "abstract": "Let $(X,τ)$ be a noncompact metrizable space and let $\\langle T_n\\rangle$ be a sequence of nonempty closed sets converging to the empty set in the sense of Kuratowski--Painlevé. The subsets of $X$ whose closures meet at most finitely many terms form a bornology $\\mathcal B_τ(\\langle T_n\\rangle)$. We give an internal criterion for inclusion between two such bornologies and recover each of them from a canonical free filter of closed sets. In fact, a nontrivial bornology is sequence-induced exactly when it has the enlargement property and its closed polar has countable character. Our principal result says that the bornology of relatively compact sets is sequence-induced precisely when $X$ is locally compact and separable; in the locally compact case the character of its closed polar is the density of $X$. In the penultimate section, we make a clever use of the sequence $T_n=\\{n\\}$ in the real line to give a negative answer to a conjecture posed in 2001 about a coarsest topology making the associated bornology a nontrivial metric bornology. Finally, we end the paper with a more general result: a coarsest such supertopology exists if and only if the original topology already makes the induced bornology a nontrivial metric bornology, in which case the coarsest supertopology is the original topology itself.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35907",
  "author_total": 2,
  "authors": [
    {
      "name": "Basile Beyer de Ryke"
    },
    {
      "name": "Gerald Beer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3sbvsf5jw1v99.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35907",
  "fulltext_url": null,
  "id": "3sbvsf5jw1v99",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35907",
  "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.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.35907",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3sbvsf5jw1v99.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35907",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35907"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Bornologies and Kuratowski-Painlevé Convergence to the Empty Set Revisited",
  "title_text": "Bornologies and Kuratowski-Painlevé Convergence to the Empty Set Revisited",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/3sbvsf5jw1v99",
  "venue": "arXiv",
  "versions": []
}