{
  "abstract": "We show that the Scott power construction does not preserve co-sobriety. For a dcpo $P$, let $\\operatorname{Fin}(P)$ be the poset of nonempty finitely generated upper sets ordered by reverse inclusion. We prove that co-sobriety of $Σ\\operatorname{Fin}(P)$ implies co-sobriety of $ΣP$. The proof adapts the point-saturation and union argument used for full compact saturated power posets; the finite step follows from a result of Heckmann and Keimel. Applying this transfer to a dcpo with a non-co-sober Scott space yields a co-sober Alexandrov space whose Scott power space is not co-sober. We also construct a countable example for which the Smyth power poset is countable. In the countable example, the same poset is co-sober under the upper Vietoris topology but not under the Scott topology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01516",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei Ji"
    },
    {
      "name": "Xiaoquan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/76z2sk9qdpvcz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01516",
  "fulltext_url": null,
  "id": "76z2sk9qdpvcz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01516",
  "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/2611.01516",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/76z2sk9qdpvcz.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01516",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01516"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A co-sober Alexandrov space whose Scott power space is not co-sober",
  "title_text": "A co-sober Alexandrov space whose Scott power space is not co-sober",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/76z2sk9qdpvcz",
  "venue": "arXiv",
  "versions": []
}