{"id":6243,"date":"2023-01-21T21:15:07","date_gmt":"2023-01-21T20:15:07","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6243"},"modified":"2023-02-20T19:19:27","modified_gmt":"2023-02-20T18:19:27","slug":"the-space-s0","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6243","title":{"rendered":"The space S0"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>S<\/em><sub>0<\/sub> is a space that occurs in Matthew de Brecht&#8217;s generalized Hurewicz theorem for quasi-Polish spaces, published in 2018.  <em>S<\/em><sub>0<\/sub> is very simple: it is an infinite countably-branching tree, and if you order it so that the root is at the top, <em>S<\/em><sub>0<\/sub> comes with the upper topology of the resulting ordering.  <em>S<\/em><sub>0<\/sub> is one of the four canonical examples of a non-quasi-Polish space (in a precise sense).  I will describe it, and I will show how closed sets and compact saturated sets in <em>S<\/em><sub>0<\/sub> can be described through certain kinds of subtrees.  With that done, we will see that <em>S<\/em><sub>0<\/sub> is sober, Choquet-complete, and completely Baire, but not locally compact, not convergence Choquet-complete, not compactly Choquet-complete, not LCS-complete, and, finally, not quasi-Polish.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6120\" data-type=\"page\" data-id=\"6120\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>S0 is a space that occurs in Matthew de Brecht&#8217;s generalized Hurewicz theorem for quasi-Polish spaces, published in 2018. S0 is very simple: it is an infinite countably-branching tree, and if you order it so that the root is at &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6243\">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],"class_list":["post-6243","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-counterexample"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6243","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=6243"}],"version-history":[{"count":2,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6243\/revisions"}],"predecessor-version":[{"id":6245,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6243\/revisions\/6245"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6243"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6243"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}