{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,20]],"date-time":"2026-06-20T02:09:26Z","timestamp":1781921366123,"version":"3.54.5"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"9","license":[{"start":{"date-parts":[[2023,7,28]],"date-time":"2023-07-28T00:00:00Z","timestamp":1690502400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2023,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The study of the sobriety of Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Soon after, Isbell gave a complete lattice with a non-sober Scott space. Based on Isbell\u2019s example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung\u2019s problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000269_inline1.png\"\/><jats:tex-math>\n$\\Sigma P\\times \\Sigma Q$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> coincides with the Scott topology of the product poset <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000269_inline2.png\"\/><jats:tex-math>\n$P\\times Q$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> if the set <jats:italic>Id<\/jats:italic>(<jats:italic>P<\/jats:italic>) and <jats:italic>Id<\/jats:italic>(<jats:italic>Q<\/jats:italic>) of all incremental ideals of posets <jats:italic>P<\/jats:italic> and <jats:italic>Q<\/jats:italic> are both countable. Based on this, it is deduced that a directed complete poset <jats:italic>P<\/jats:italic> has a sober Scott space, if <jats:italic>Id<\/jats:italic>(<jats:italic>P<\/jats:italic>) is countable and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000269_inline3.png\"\/><jats:tex-math>\n$\\Sigma P$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is coherent and well filtered. In particular, every complete lattice <jats:italic>L<\/jats:italic> with <jats:italic>Id<\/jats:italic>(<jats:italic>L<\/jats:italic>) countable has a sober Scott space.<\/jats:p>","DOI":"10.1017\/s0960129523000269","type":"journal-article","created":{"date-parts":[[2023,7,28]],"date-time":"2023-07-28T10:49:42Z","timestamp":1690541382000},"page":"809-831","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":5,"title":["Not every countable complete distributive lattice is sober"],"prefix":"10.1017","volume":"33","author":[{"given":"Hualin","family":"Miao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6124-395X","authenticated-orcid":false,"given":"Xiaoyong","family":"Xi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4314-7489","authenticated-orcid":false,"given":"Qingguo","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongsheng","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2023,7,28]]},"reference":[{"key":"S0960129523000269_ref9","unstructured":"Jung, A. (2018). Four dcpos, a theorem, and an open problem, an invited talk in Hunan University, Hunan Province, China, 1 December 2018."},{"key":"S0960129523000269_ref3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1969-0251026-X"},{"key":"S0960129523000269_ref10","first-page":"357","article-title":"The duality of continuous posets","volume":"5","author":"Lawson","year":"1979","journal-title":"Houston Journal of Mathematics"},{"key":"S0960129523000269_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2017.06.002"},{"key":"S0960129523000269_ref12","doi-asserted-by":"publisher","DOI":"10.4064\/fm704-4-2020"},{"key":"S0960129523000269_ref1","volume-title":"New Mathematical Monographs","volume":"22","author":"Goubault-Larrecq","year":"2013"},{"key":"S0960129523000269_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2020.107540"},{"key":"S0960129523000269_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0089907"},{"key":"S0960129523000269_ref5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1982-0656096-4"},{"key":"S0960129523000269_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2016.06.011"},{"key":"S0960129523000269_ref7","unstructured":"Jia, X. (2018). Meet-Continuity and Locally Compact Sober Dcpos. Phd thesis, University of Birmingham."},{"key":"S0960129523000269_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0089911"},{"key":"S0960129523000269_ref2","volume-title":"Encyclopedia of Mathematics and its Applications","volume":"93","author":"Gierz","year":"2003"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129523000269","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,1]],"date-time":"2023-11-01T11:21:33Z","timestamp":1698837693000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129523000269\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,28]]},"references-count":13,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["S0960129523000269"],"URL":"https:\/\/doi.org\/10.1017\/s0960129523000269","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,28]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}