Scott topologies on meet-continuous domains

@misc{xu2026scott,
  title = {{Scott topologies on meet-continuous domains}},
  author = {Xiaoquan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21274},
  eprint = {2609.21274},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21274},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Scott topologies on meet-continuous domains
AU  - Xiaoquan Xu
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21274
UR  - https://arxiv.org/abs/2609.21274
AB  - We study Scott products and sobriety of countable meet-continuous domains, meaning meet-continuous dcpos without any additional continuity or least-element assumption. Using the complete-lattice test-family theorem of Xu and Ji, we prove finite-product equality for those domains that are L-dcpos, and for the weaker class whose principal ideals have suprema of all nonempty subsets. For an arbitrary family of nonempty countable meet-continuous L-dcpos, we prove that the Scott topology on the order product equals the product of the factor Scott topologies if and only if only finitely many factors lack a least element. We also establish sobriety under bounded completeness and under additional common-upper-bound conditions. Assuming square-product equality, sobriety is characterized by Scott closedness of common-upper-bound sections associated with irreducible Scott-closed sets, with an equivalent sequential formulation in the countable case. Two extraction lemmas extend to meet-semilattice dcpos. The finite-product and sobriety questions for general countable meet-continuous domains remain unresolved here.
ER  -