{
  "abstract": "Let $\\mathcal I$ be an admissible ideal on $\\mathbb N$. Zhou, Liu, Liu, and Lin asked whether every regular space with a $σ$-hereditarily closure-preserving $\\mathcal I$-$sn$-network is $\\mathcal I$-$sn$-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a local countability theorem. If $y_n\\to_{\\mathcal I}x$ and $\\{n\\mid y_n\\ne x\\}\\in\\mathcal I^+$, then every point-discrete family of $\\mathcal I$-sequential neighborhoods of $x$ is countable. In a $T_1$ space, this applies in particular to every HCP family. If no $\\mathcal I$-convergent sequence to $x$ has $\\mathcal I$-positive support away from $x$, then $\\{x\\}$ is an $\\mathcal I$-sequential neighborhood. It follows that every space with a $σ$-point-discrete $\\mathcal I$-$sn$-network is $\\mathcal I$-$snf$-countable. This conclusion, together with the characterization theorems of Ge and of Zhou, Liu, Liu, and Lin, settles Problem 5.2. The countability bound cannot in general be strengthened to finiteness. A countable Rudin--Keisler antichain of free ultrafilters gives an admissible ideal for which local HCP-finiteness fails. Every $P^{-}$ ideal has local HCP-finiteness. Preservation theorems for Fubini products, heterogeneous Fubini sums under cross-row Katětov absorption, and suitable increasing unions extend local HCP-finiteness beyond the $P^{-}$ class. They apply to the finite Fubini powers of $\\mathrm{Fin}$, Katětov's first limit-stage ideal $\\mathrm{Fin}^ω$ and its finite successors, the tree-derived rank-$ω$ ideal $\\mathcal H_{<ω}$, and the inductive-limit ideal $\\mathrm{Fin}_ω$. Every listed example other than $\\mathrm{Fin}$ fails $P^{-}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30317",
  "author_total": 1,
  "authors": [
    {
      "name": "Xing-Yu Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fggp06d69c52d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30317",
  "fulltext_url": null,
  "id": "fggp06d69c52d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30317",
  "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.GN"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214596439",
  "pdf_url": "https://arxiv.org/pdf/2609.30317",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fggp06d69c52d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30317",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30317"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "$σ$-hereditarily closure-preserving $\\mathcal I$-$sn$-networks, $\\mathcal I$-$sn$-metrizability, and local HCP-finiteness",
  "title_text": "σ-hereditarily closure-preserving ℐ-sn-networks, ℐ-sn-metrizability, and local HCP-finiteness",
  "type": "preprint",
  "updated_at": "2026-11-05T15:21:00Z",
  "url": "https://research.draconitic.com/paper/fggp06d69c52d",
  "venue": "arXiv",
  "versions": []
}