{
  "abstract": "We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset $A$ of a $T_1$-space $(X,τ)$, we consider the topology consisting of the co-finite open sets together with the $τ$-open sets disjoint from $A$. We determine its $k$-irreducible compact subsets and derive a criterion that produces two compact co-sober spaces whose product contains a non-singleton $k$-irreducible compact saturated subset of the diagonal. Applying the criterion to a specific Kovář--van Douwen space gives two countable compact Fréchet--Urysohn $T_1$ anti-Hausdorff co-sober spaces whose product is not co-sober. A further construction yields a countable compact Fréchet--Urysohn $T_1$ anti-Hausdorff co-sober space whose square is not co-sober. The failure of product closure also gives an alternative proof of the known non-reflectivity of co-sober spaces in the category of $T_0$-spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14434",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoquan Xu"
    },
    {
      "name": "Wei Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dryhvzytrmnce.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14434",
  "fulltext_url": null,
  "id": "dryhvzytrmnce",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14434",
  "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": "W7221785261",
  "pdf_url": "https://arxiv.org/pdf/2610.14434",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dryhvzytrmnce.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14434",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14434"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Products of co-sober spaces need not be co-sober",
  "title_text": "Products of co-sober spaces need not be co-sober",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/dryhvzytrmnce",
  "venue": "arXiv",
  "versions": []
}