{"id":2039,"date":"2019-09-24T11:08:36","date_gmt":"2019-09-24T09:08:36","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2039"},"modified":"2023-03-20T11:41:29","modified_gmt":"2023-03-20T10:41:29","slug":"well-filterifications","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2039","title":{"rendered":"Well-filterifications"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Xiaodong Jia once asked the following <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1707\">question<\/a>: is every core-compact, well-filtered space automatically locally compact?  The question was solved positively this year by J. Lawson and X. Xi.  I originally planned to try and explain their result.  Even more recently, X. Xu, Ch. Shen, X. Xi and D. Zhao found a simpler solution, and I have changed my plans.  My new plan for this time, and next time, is to explain what they have done.  This time, we will concentrate on <em>well-filterifications<\/em> of topological space, which are just like sobrifications except &#8216;sober&#8217; is replaced by &#8216;well-filtered T<sub>0<\/sub>&#8216;.  Building one is not completely obvious.  Xu, Shen, Xi and Zhao show that the will-filterification of <em>X<\/em> can be defined as its set of closed <em>WD sets<\/em>, a new notion that is intermediate between directed sets and irreducible sets; the proof also relies on a  refinement of R. Heckmann and K. Keimel&#8217;s topological version of Rudin&#8217;s Lemma, which is interesting in its own right.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2014\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Xiaodong Jia once asked the following question: is every core-compact, well-filtered space automatically locally compact? The question was solved positively this year by J. Lawson and X. Xi. I originally planned to try and explain their result. Even more recently, &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2039\">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":[54],"class_list":["post-2039","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-well-filtered-space"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2039","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=2039"}],"version-history":[{"count":3,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2039\/revisions"}],"predecessor-version":[{"id":2046,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2039\/revisions\/2046"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2039"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2039"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}