{
  "abstract": "For a finite set $S$ with $\\lvert S\\rvert = N$, the number of families $\\mathcal{B} \\subseteq \\mathcal{P}(S)$ that are topological bases is $\\#(N) = \\sum_{\\mathcal{T} \\in \\operatorname{Top}(S)} 2^{\\lvert\\mathcal{T}\\rvert - \\lvert\\mathcal{M}_{\\mathcal{T}}\\rvert}$, where $\\mathcal{M}_{\\mathcal{T}}$ is the canonical minimal basis of minimal open neighborhoods. The identity is proved in Lean 4 / Mathlib (`CARDB.lean`): bases generating $\\mathcal{T}$ are exactly the sets with $\\mathcal{M}_{\\mathcal{T}} \\subseteq \\mathcal{B} \\subseteq \\mathcal{T}$. The small-$N$ table and the discrete-dominance sandwich are proved in `CARDB/SmallN.lean` and `CARDB/Asymptotics.lean`.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14102",
  "author_total": 1,
  "authors": [
    {
      "name": "Lars Warren Ericson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2950x7ef1c5j0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14102",
  "fulltext_url": null,
  "id": "2950x7ef1c5j0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14102",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
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      "arxiv",
      "math.CO"
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      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
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  ],
  "openalex": "W7213351245",
  "pdf_url": "https://arxiv.org/pdf/2609.14102",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2950x7ef1c5j0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14102",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14102"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the Number of Distinct Topological Bases of a Finite Set of Size $N$",
  "title_text": "On the Number of Distinct Topological Bases of a Finite Set of Size N",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/2950x7ef1c5j0",
  "venue": "arXiv",
  "versions": []
}