{
  "abstract": "Xu asked whether the Hoare power space of a co-sober space, and in particular of a $T_2$-space or a metric space, must be co-sober. We give a negative answer. He and Zhao recently constructed a countable $T_{1}$ space $X$ whose lattice of open sets has a non-sober Scott topology. We observe that their topology has a countable subbase. The classical Ponomarev construction then yields an open continuous surjection from a countable zero-dimensional metrizable space $M$ onto $X$. Direct and inverse images then make the Scott space $Σ\\mathcal{O}(X)$ a retract of $Σ\\mathcal{O}(M)$, so $Σ\\mathcal{O}(M)$ is not sober. Finally, we show that, for every $T_{0}$-space $Y$, the Hoare power space $\\PH(Y)$ is co-sober if and only if the Scott space $Σ\\mathcal{O}(Y)$ is sober. Consequently, $\\PH(M)$ is not co-sober. Therefore, the same example answers negatively a question of He and Zhao about Scott sobriety of the lattices of open sets of countable Hausdorff spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05868",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoquan Xu"
    },
    {
      "name": "Wei Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a4mk0meqfvrfa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05868",
  "fulltext_url": null,
  "id": "a4mk0meqfvrfa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05868",
  "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": "W7220661261",
  "pdf_url": "https://arxiv.org/pdf/2610.05868",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a4mk0meqfvrfa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05868",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05868"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A countable metric space whose Hoare power space is not co-sober",
  "title_text": "A countable metric space whose Hoare power space is not co-sober",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/a4mk0meqfvrfa",
  "venue": "arXiv",
  "versions": []
}