{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:57:25Z","timestamp":1747173445905,"version":"3.40.5"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2022,6,27]],"date-time":"2022-06-27T00:00:00Z","timestamp":1656288000000},"content-version":"unspecified","delay-in-days":177,"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":[[2022,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the <jats:italic>d<\/jats:italic>-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline1.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and <jats:italic>d<\/jats:italic>-spaces. In addition, a characterization for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline2.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-well-filtered spaces is provided via <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline3.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-sets. We also discuss the relationship between <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline4.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-well-filtered spaces and <jats:italic>H<\/jats:italic>-sober spaces considered by Xu. We show that the category of complete <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline5.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-well-filtered spaces is a full reflective subcategory of the category of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline6.png\"\/><jats:tex-math>\n$T_0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> spaces with continuous mappings. For each HM-system <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline7.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> that has a designated property, we show that a <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline8.png\"\/><jats:tex-math>\n$T_0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> space <jats:italic>X<\/jats:italic> is <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline9.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-well-filtered if and only if its Smyth power space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline10.png\"\/><jats:tex-math>\n$P_s(X)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129522000196_inline11.png\"\/><jats:tex-math>\n$\\Psi$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-well-filtered.<\/jats:p>","DOI":"10.1017\/s0960129522000196","type":"journal-article","created":{"date-parts":[[2022,6,27]],"date-time":"2022-06-27T09:43:51Z","timestamp":1656323031000},"page":"111-124","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Hofmann-Mislove type definitions of non-Hausdorff spaces"],"prefix":"10.1017","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5838-0438","authenticated-orcid":false,"given":"Chong","family":"Shen","sequence":"first","affiliation":[]},{"given":"Xiaoyong","family":"Xi","sequence":"additional","affiliation":[]},{"given":"Xiaoquan","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Dongsheng","family":"Zhao","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,6,27]]},"reference":[{"key":"S0960129522000196_ref8","first-page":"125","volume":"105","author":"Hoffmann","year":"1981"},{"key":"S0960129522000196_ref16","first-page":"4","article-title":"On open well-filtered spaces","volume":"16","author":"Shen","year":"2020","journal-title":"Logical Methods in Computer Science"},{"key":"S0960129522000196_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2016.06.011"},{"key":"S0960129522000196_ref14","unstructured":"Schalk, A. 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