{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T12:40:08Z","timestamp":1745239208933,"version":"3.40.4"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T00:00:00Z","timestamp":1745193600000},"content-version":"unspecified","delay-in-days":110,"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":[[2025]]},"abstract":"<jats:title>Abstract<\/jats:title>\n\t  <jats:p>A topological space has a domain model if it is homeomorphic to the maximal point space <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline3.png\"\/>\n\t\t<jats:tex-math>\n$\\mbox{Max}(P)$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> of a domain <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline4.png\"\/>\n\t\t<jats:tex-math>\n$P$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>. Lawson proved that every Polish space <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline5.png\"\/>\n\t\t<jats:tex-math>\n$X$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> has an <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline6.png\"\/>\n\t\t<jats:tex-math>\n$\\omega$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-domain model <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline7.png\"\/>\n\t\t<jats:tex-math>\n$P$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> and for such a model <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline8.png\"\/>\n\t\t<jats:tex-math>\n$P$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>, <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline9.png\"\/>\n\t\t<jats:tex-math>\n$\\mbox{Max}(P)$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> is a <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline10.png\"\/>\n\t\t<jats:tex-math>\n$G_{\\delta }$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-set of the Scott space of <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline11.png\"\/>\n\t\t<jats:tex-math>\n$P$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>. Martin (2003) then asked whether it is true that for every <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline12.png\"\/>\n\t\t<jats:tex-math>\n$\\omega$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-domain <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline13.png\"\/>\n\t\t<jats:tex-math>\n$Q$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>, <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline14.png\"\/>\n\t\t<jats:tex-math>\n$\\mbox{Max}(Q)$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> is <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline15.png\"\/>\n\t\t<jats:tex-math>\n$G_{\\delta }$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-set of the Scott space of <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline16.png\"\/>\n\t\t<jats:tex-math>\n$Q$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>. In this paper, we give a negative answer to Martin\u2019s long-standing open problem by constructing a counterexample. The counterexample here actually shows that the answer is no even for <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline17.png\"\/>\n\t\t<jats:tex-math>\n$\\omega$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-algebraic domains. In addition, we also construct an <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline18.png\"\/>\n\t\t<jats:tex-math>\n$\\omega$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-ideal domain <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline19.png\"\/>\n\t\t<jats:tex-math>\n$\\widetilde{Q}$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> for the constructed <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline20.png\"\/>\n\t\t<jats:tex-math>\n$Q$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> such that their maximal point spaces are homeomorphic. Therefore, <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline21.png\"\/>\n\t\t<jats:tex-math>\n$\\textrm{Max}(Q)$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> is a <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline22.png\"\/>\n\t\t<jats:tex-math>\n$G_\\delta$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>-set of the Scott space of the new model <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000392_inline23.png\"\/>\n\t\t<jats:tex-math>\n$\\widetilde{Q}$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> .<\/jats:p>","DOI":"10.1017\/s0960129524000392","type":"journal-article","created":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T12:13:22Z","timestamp":1745237602000},"update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["The set of maximal points of an \n$\\boldsymbol{\\omega}$\n-domain need not be a \n$\\boldsymbol{G}_{\\boldsymbol{\\delta}}$\n-set"],"prefix":"10.1017","volume":"35","author":[{"given":"Gaolin","family":"Li","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5838-0438","authenticated-orcid":false,"given":"Chong","family":"Shen","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6460-6990","authenticated-orcid":false,"given":"Kaiyun","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6124-395X","authenticated-orcid":false,"given":"Xiaoyong","family":"Xi","sequence":"additional","affiliation":[]},{"given":"Dongsheng","family":"Zhao","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2025,4,21]]},"reference":[{"key":"S0960129524000392_ref4","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1017\/S0960129597002363","article-title":"Spaces of maximal points","volume":"7","author":"Lawson","year":"1997","journal-title":"Mathematical Structures in Computer Science"},{"key":"S0960129524000392_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004116000888"},{"key":"S0960129524000392_ref1","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129597002338"},{"key":"S0960129524000392_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(02)00700-4"},{"key":"S0960129524000392_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139524438"},{"key":"S0960129524000392_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(02)00698-9"},{"key":"S0960129524000392_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511542725"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129524000392","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T12:13:22Z","timestamp":1745237602000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129524000392\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"references-count":7,"alternative-id":["S0960129524000392"],"URL":"https:\/\/doi.org\/10.1017\/s0960129524000392","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025]]},"assertion":[{"value":"\u00a9 Yancheng Teachers University, 2025. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}],"article-number":"e5"}}