{"id":2324,"date":"2020-03-20T18:18:31","date_gmt":"2020-03-20T17:18:31","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2324"},"modified":"2023-03-20T11:34:58","modified_gmt":"2023-03-20T10:34:58","slug":"x-jias-well-filtered-non-sober-dcpo","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2324","title":{"rendered":"X. Jia&#8217;s well-filtered, non-sober dcpo"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">[Business as usual, despite all viruses!]  Peter Johnstone once showed the existence of a dcpo <strong>J<\/strong> that is not sober in its Scott topology.  That dcpo is not well-filtered either.  Is there a dcpo that is not sober but <em>is<\/em> well-filtered?  That is true, and the first one who found an example is Hui Kou.  Since then, Xi and Zhao have also given another example, and I would like to describe another example of such a dcpo, due to Xiaodong Jia in his PhD thesis.  Both Xi and Zhao&#8217;s example and Jia&#8217;s example are pretty simple spaces, but X. Jia&#8217;s example is countable.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2259\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[Business as usual, despite all viruses!] Peter Johnstone once showed the existence of a dcpo J that is not sober in its Scott topology. That dcpo is not well-filtered either. Is there a dcpo that is not sober but is &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2324\">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-2324","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\/2324","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=2324"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2324\/revisions"}],"predecessor-version":[{"id":2325,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2324\/revisions\/2325"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2324"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2324"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2324"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}