{"id":2616,"date":"2020-06-20T10:15:27","date_gmt":"2020-06-20T08:15:27","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2616"},"modified":"2023-03-20T11:33:25","modified_gmt":"2023-03-20T10:33:25","slug":"zhao-xi-and-chens-non-sober-well-filtered-dcpo","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2616","title":{"rendered":"Zhao, Xi and Chen&#8217;s well-filtered, non-sober dcpo"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">There are several known examples of dcpos that are well-filtered, but not sober, and I have already mentioned one due to <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2259\">Xiaodong Jia<\/a>. I would like to explain another one, due to Dongsheng Zhao, Xiaoyong Xi, and Yixiang Chen. This is a very simple modification of Johnstone&#8217;s non-sober dcpo <strong>J<\/strong>. Contrarily to Xiaodong Jia&#8217;s dcpo (and to <strong>J<\/strong>), it is uncountable, but it may be easier to see why it must be well-filtered: everything mostly boils down to a cardinality argument, or rather, as I will argue, to the properties of so-called regular ordinals.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2507\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>There are several known examples of dcpos that are well-filtered, but not sober, and I have already mentioned one due to Xiaodong Jia. I would like to explain another one, due to Dongsheng Zhao, Xiaoyong Xi, and Yixiang Chen. This &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2616\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","footnotes":""},"categories":[1],"tags":[18,55,54],"class_list":["post-2616","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-counterexample","tag-sober-space","tag-well-filtered-space"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2616","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2616"}],"version-history":[{"count":2,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2616\/revisions"}],"predecessor-version":[{"id":2624,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2616\/revisions\/2624"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2616"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2616"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2616"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}