{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T03:08:06Z","timestamp":1767236886156},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2015,5,12]],"date-time":"2015-05-12T00:00:00Z","timestamp":1431388800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2017,5]]},"abstract":"<jats:p>A topological space <jats:italic>X<\/jats:italic> is called <jats:italic>well-filtered<\/jats:italic> if for any filtered family <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0960129515000171_inline1\" \/><jats:tex-math>$\\mathcal{F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of compact saturated sets and an open set <jats:italic>U<\/jats:italic>, \u2229 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0960129515000171_inline1\" \/><jats:tex-math>$\\mathcal{F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> \u2286 <jats:italic>U<\/jats:italic> implies <jats:italic>F<\/jats:italic> \u2286 <jats:italic>U<\/jats:italic> for some <jats:italic>F<\/jats:italic> \u2208 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0960129515000171_inline1\" \/><jats:tex-math>$\\mathcal{F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Every sober space is well-filtered and the converse is not true. A dcpo (directed complete poset) is called well-filtered if its Scott space is well-filtered. In 1991, Heckmann asked whether every <jats:italic>U<jats:sub>K<\/jats:sub><\/jats:italic>-admitting (the same as well-filtered) dcpo is sober. In 2001, Kou constructed a counterexample to give a negative answer. In this paper, for each <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub> space <jats:italic>X<\/jats:italic> we consider a dcpo <jats:italic>D<\/jats:italic>(<jats:italic>X<\/jats:italic>) whose maximal point space is homeomorphic to <jats:italic>X<\/jats:italic> and prove that <jats:italic>X<\/jats:italic> is well-filtered if and only if <jats:italic>D<\/jats:italic>(<jats:italic>X<\/jats:italic>) is well-filtered. The main result proved here enables us to construct new well-filtered dcpos that are not sober (only one such example is known by now). A space will be called K-closed if the intersection of every filtered family of compact saturated sets is compact. Every well-filtered space is K-closed. Some similar results on K-closed spaces are also proved.<\/jats:p>","DOI":"10.1017\/s0960129515000171","type":"journal-article","created":{"date-parts":[[2015,7,22]],"date-time":"2015-07-22T22:40:08Z","timestamp":1437604808000},"page":"507-515","source":"Crossref","is-referenced-by-count":25,"title":["Well-filtered spaces and their dcpo models"],"prefix":"10.1017","volume":"27","author":[{"given":"XIAOYONG","family":"XI","sequence":"first","affiliation":[]},{"given":"DONGSHENG","family":"ZHAO","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2015,5,12]]},"reference":[{"key":"S0960129515000171_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-55511-0_14"},{"key":"S0960129515000171_ref5","first-page":"41","article-title":"Well-filtered DCPOS need not be sober","volume":"1","author":"Kou","year":"2010","journal-title":"Domains and Processes, Semantic Structures in Computation"},{"key":"S0960129515000171_ref1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511542725"},{"key":"S0960129515000171_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0089911"},{"key":"S0960129515000171_ref6","unstructured":"Zhao D. (2009). Poset models of topological spaces. In: Proceeding of International Conference on Quantitative Logic and Quantification of Software, Global-Link Publisher 229\u2013238."},{"key":"S0960129515000171_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2013.09.015"},{"key":"S0960129515000171_ref7","unstructured":"Zhao D. and Xi X. (2014). Dcpo-models of T1-spaces (Under consideration for publication by Mathematical Proceedings of the Cambridge Philosophical Society)."}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129515000171","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,17]],"date-time":"2019-04-17T20:04:19Z","timestamp":1555531459000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129515000171\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,12]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,5]]}},"alternative-id":["S0960129515000171"],"URL":"https:\/\/doi.org\/10.1017\/s0960129515000171","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,5,12]]}}}