{"id":2088,"date":"2019-10-20T09:07:12","date_gmt":"2019-10-20T07:07:12","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2088"},"modified":"2023-03-20T11:41:20","modified_gmt":"2023-03-20T10:41:20","slug":"core-compactwell-filtered-t0sober-locally-compact","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2088","title":{"rendered":"Core-compact+well-filtered T0=sober locally compact"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2014\">Last time<\/a>, I motivated the construction of the well-filterification <strong>Wf<\/strong>(<em>X<\/em>) of a space <em>X<\/em> of X. Xu, Ch. Shen, X. Xi and D. Zhao by saying that it was needed to understand their proof of the fact that every core-compact well-filtered T<sub>0<\/sub> space is sober, and hence also locally compact.  This solves a <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1707\">question<\/a> asked by X. Jia, and first solved positively by J. Lawson and X. Xi.  I then realized that their proof contains a (somewhat concealed) gem; polishing it reveals an interesting new variant of the Hofmann-Mislove theorem, which applies to well-filtered spaces rather than to sober spaces, and rests on a funny countability assumption.  We will see that the latter is satisfied in all core-compact spaces, and this will allow us to show that all core-compact well-filtered T<sub>0<\/sub> spaces are locally compact, hence sober.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2067\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Last time, I motivated the construction of the well-filterification Wf(X) of a space X of X. Xu, Ch. Shen, X. Xi and D. Zhao by saying that it was needed to understand their proof of the fact that every core-compact &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2088\">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":[30,55,54],"class_list":["post-2088","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-core-compactness","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\/2088","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=2088"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2088\/revisions"}],"predecessor-version":[{"id":2090,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2088\/revisions\/2090"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2088"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2088"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2088"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}