{
  "abstract": "We prove that the Scott topology commutes with arbitrary products of countably presented frames and that every such product has a sober Scott space. In particular, this holds for arbitrary products of countable complete Heyting algebras. For a family of consonant spaces, Scott-product compatibility of their open-set lattices is equivalent to consonance of their topological sum. Countable generation does not suffice: a countably generated spatial frame can have a non-sober Scott space and fail the product identity for its square. We also show that the cardinal spectra of Scott non-sober frames and spatial frames are upward closed and, under the Continuum Hypothesis, consist of all uncountable cardinals. Finally, every Artinian $T_0$ web space is a $B$-space; if it is also a $d$-space, it carries the Scott topology of an algebraic dcpo. Consequently, every Artinian meet-continuous dcpo is algebraic. This yields a dichotomy theorem: a dcpo with a non-sober Scott space must fail meet continuity or Artinianity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18032",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaoquan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4enpj1kdd36dg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18032",
  "fulltext_url": null,
  "id": "4enpj1kdd36dg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18032",
  "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.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7213513159",
  "pdf_url": "https://arxiv.org/pdf/2609.18032",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4enpj1kdd36dg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18032",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18032"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Scott topologies on products of countable complete Heyting algebras",
  "title_text": "Scott topologies on products of countable complete Heyting algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/4enpj1kdd36dg",
  "venue": "arXiv",
  "versions": []
}