Scott topologies on products of countable complete Heyting algebras

@misc{xu2026scott,
  title = {{Scott topologies on products of countable complete Heyting algebras}},
  author = {Xiaoquan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18032},
  eprint = {2609.18032},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18032},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Scott topologies on products of countable complete Heyting algebras
AU  - Xiaoquan Xu
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18032
UR  - https://arxiv.org/abs/2609.18032
AB  - 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.
ER  -