{
  "abstract": "Kernel retracts of co-sober spaces need not be co-sober, even when the ambient space is countably infinite and compact. We construct a co-sober $T_0$-space $Y$ and a continuous idempotent map $k\\colon Y\\to Y$ satisfying $k(y)\\leq_Y y$ in the specialization order, whose image with the subspace topology is not co-sober. The construction starts from a countable compact $T_1$ KC anti-Hausdorff space of van Douwen type and adjoins an infinite chain. We classify the $k$-irreducible compact saturated subsets of the extension and show that each is a point saturation. The kernel operator collapses the added chain to a distinguished point while fixing the original space, thereby recovering the non-co-sober seed as its image. This answers negatively two questions of Xu and Zhao, concerning continuous kernel operators and arbitrary retracts, and contrasts with the known preservation of co-sobriety by continuous closure operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01531",
  "author_total": 4,
  "authors": [
    {
      "name": "Xiaoquan Xu"
    },
    {
      "name": "Fengshuo Xu"
    },
    {
      "name": "Haitao Ji"
    },
    {
      "name": "Dong Qiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cd4rf6hwj76za.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01531",
  "fulltext_url": null,
  "id": "cd4rf6hwj76za",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01531",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01531",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cd4rf6hwj76za.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01531",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01531"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Kernel retracts of co-sober spaces",
  "title_text": "Kernel retracts of co-sober spaces",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/cd4rf6hwj76za",
  "venue": "arXiv",
  "versions": []
}