{"id":1582,"date":"2018-10-23T19:50:45","date_gmt":"2018-10-23T17:50:45","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1582"},"modified":"2020-04-03T10:38:45","modified_gmt":"2020-04-03T08:38:45","slug":"project-limits-of-topological-spaces-iii-finishing-the-proof-of-steenrods-theorem","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1582","title":{"rendered":"Projective limits of topological spaces III: finishing the proof of Steenrod&#8217;s theorem"},"content":{"rendered":"<p><a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1571\">Last time<\/a>, we embarked on proving that the projective limit of a projective system of compact sober (resp., and non-empty) spaces is compact and sober (resp., and non-empty), a theorem that Fujiwara and Kato call Steenrod&#8217;s Theorem. &nbsp;However, instead, we merely proved that a&nbsp;projective limit of a projective system of non-empty compact sober spaces is non-empty. &nbsp;Do not despair: this is the essential argument in the proof of Steenrod&#8217;s Theorem, which we complete this month. &nbsp;Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1559\">full post<\/a>.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Last time, we embarked on proving that the projective limit of a projective system of compact sober (resp., and non-empty) spaces is compact and sober (resp., and non-empty), a theorem that Fujiwara and Kato call Steenrod&#8217;s Theorem. &nbsp;However, instead, we &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1582\">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":[],"class_list":["post-1582","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/1582","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=1582"}],"version-history":[{"count":3,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/1582\/revisions"}],"predecessor-version":[{"id":1594,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/1582\/revisions\/1594"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1582"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1582"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1582"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}