{
  "abstract": "We exhibit sober spaces $X$ and $Y$ such that the function space $[X \\to Y]_K$ of continuous maps, equipped with the compact-open topology, is not sober; the codomain can be chosen to be the Sierpiński space $S$. Our construction builds on Isbell's complete lattice $M$, whose Scott space $ΣM$ is non-sober. Let $H = Q(M)$ be the set of nonempty compact saturated subsets of $ΣM$, ordered by reverse inclusion. Since $H$ is well known to be a spatial frame, the space $X = \\mathrm{pt}(H)$ is sober with $\\mathcal{O}(X) \\cong H$. The key technical result is that sobriety of the space $H$ would entail sobriety of the original Scott space $ΣM$. This contradicts Isbell's example, and the desired counterexample follows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36551",
  "author_total": 2,
  "authors": [
    {
      "name": "Bei Liu"
    },
    {
      "name": "Qingguo Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/58wcveeza9qhb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36551",
  "fulltext_url": null,
  "id": "58wcveeza9qhb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36551",
  "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": "W7224097638",
  "pdf_url": "https://arxiv.org/pdf/2610.36551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/58wcveeza9qhb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Function spaces with compact-open topology need not be sober",
  "title_text": "Function spaces with compact-open topology need not be sober",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/58wcveeza9qhb",
  "venue": "arXiv",
  "versions": []
}