{
  "abstract": "Let $(X,τ)$ be metrizable and let $a\\in X$. We give a constructive account of metrizable topologies $σ\\subseteqτ$ that agree with $τ$ on $X\\setminus\\{a\\}$. Applying Hausdorff's classical metric collapse construction, for every noncompact $(X,τ)$ and every compatible metric $d$ we obtain a strict coarsening with a metric $p\\le d$ that agrees with $d$ on a common neighborhood of each point other than $a$. A prescribed countably infinite closed discrete set $\\{x_n:n\\in\\N\\}\\subseteq X\\setminus\\{a\\}$ can be made to satisfy $p(a,x_n)\\leλ_n$ for any positive null sequence $(λ_n)$. The resulting metric is greatest among the metrics dominated by $d$ that satisfy these bounds, and is complete whenever $d$ is complete. We exhibit its realization as a classical metric quotient. We also represent all localized metrizable coarsenings by continuous scalar gauges using a standard cone metric. Inclusion is expressed by the cofinal comparison of sublevel sets familiar from extension-trace theory, while pointwise maximum and minimum realize finite joins and meets. A closed-discrete criterion detects strictness. Standard preservation results for Borel structure, complete metrizability, and Polishness, together with function-space and local-field examples, complete the account.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13827",
  "author_total": 2,
  "authors": [
    {
      "name": "Ahmad ja'afari kalvan"
    },
    {
      "name": "Ehsan Shahoseini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4cxq7dfe2qp9t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13827",
  "fulltext_url": null,
  "id": "4cxq7dfe2qp9t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13827",
  "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",
      "math.GN"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7213258366",
  "pdf_url": "https://arxiv.org/pdf/2609.13827",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4cxq7dfe2qp9t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13827",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13827"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation",
  "title_text": "One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4cxq7dfe2qp9t",
  "venue": "arXiv",
  "versions": []
}