{
  "abstract": "We investigate how order structure controls sequential convergence in the lower topology of a poset and how this, in turn, governs topological duality and co-sobriety of Scott spaces. We first give a purely order-theoretic characterization of lower convergence. For complete lattices this yields a lower-limsup formula, and on complete chains the lower limsup coincides with the usual lattice limsup. We also characterize first countability of the lower topology at a point by the existence of a countable coinitial subset outside its principal ideal. We prove that every compact saturated subset of a Scott space is sequentially closed in the lower topology and deduce that, for every poset, the lower topology and the co-compact (de Groot dual) topology of its Scott space have exactly the same convergent sequences and limits. For coherent Scott spaces we identify the lower closure of a $k$-irreducible compact saturated set with its double-cone closure and obtain an infimum criterion for co-sobriety. Consequently, every bounded-complete dcpo with sequential lower topology has a co-sober Scott space. In particular, every countable bounded-complete dcpo, and hence every countable complete lattice, has a co-sober Scott space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01487",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoquan Xu"
    },
    {
      "name": "Wei Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cq7exg4a80wac.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01487",
  "fulltext_url": null,
  "id": "cq7exg4a80wac",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01487",
  "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/2611.01487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cq7exg4a80wac.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sequential convergence in the lower topology and co-sobriety of Scott spaces",
  "title_text": "Sequential convergence in the lower topology and co-sobriety of Scott spaces",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/cq7exg4a80wac",
  "venue": "arXiv",
  "versions": []
}