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Johnstone’e non-sober dcpo and extensions
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Type
Article
Citation
Zhao, D. (2025). Johnstone’e non-sober dcpo and extensions. International Journal of Topology, 2(1), Article 3. https://doi.org/10.3390/ijt2010003
Abstract
One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in many other parts of domain theory and more properties of it have been uncovered. In this survey paper, we first collect and prove the major properties (some of which are new as far as we know) of Johnstone’s dcpo. We then propose a general method of constructing a dcpo from given posets and prove some properties. Some problems are posed for further investigation. This paper can serve as a relatively complete resource on Johnstone’s dcpo.
Date Issued
2025
Publisher
MDPI
Journal
International Journal of Topology
Description
The open access publication is available at https://doi.org/10.3390/ijt2010003