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PreprintCC BY 4.0arXiv

Sobriety of Scott topologies under countability conditions

  • Zhengmao He
  • Zhaochen Dong
  • Kaiyun Wang
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Abstract

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

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Cite

BibTeXRIS
BibTeX
@misc{he2026sobriety,
  title = {{Sobriety of Scott topologies under countability conditions}},
  author = {Zhengmao He and Zhaochen Dong and Kaiyun Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24001},
  eprint = {2609.24001},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24001},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sobriety of Scott topologies under countability conditions
AU  - Zhengmao He
AU  - Zhaochen Dong
AU  - Kaiyun Wang
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.24001
UR  - https://arxiv.org/abs/2609.24001
AB  - In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.
ER  -