{"id":1571,"date":"2018-09-23T16:11:49","date_gmt":"2018-09-23T14:11:49","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1571"},"modified":"2020-04-03T10:38:50","modified_gmt":"2020-04-03T08:38:50","slug":"projective-limits-of-topological-space-ii-steenrods-theorem","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1571","title":{"rendered":"Projective limits of topological space II: Steenrod&#8217;s theorem"},"content":{"rendered":"<p><a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1534\">Last time<\/a>, I explained some of the strange things that happen with projective limits of topological spaces: they can be empty, even if all the spaces in the given projective system are non-empty and all bonding maps are surjective, and they can fail to be compact, even if all the spaces in the projective system are compact.<\/p>\n<p>Steenrod&#8217;s Theorem (as Fujiwara and Kato call it) shows that all those pathologies disappear if we work with compact sober spaces. \u00a0This rests on a lemma, according to which projective limits of non-empty compact sober spaces are non-empty, which is the subject of this month&#8217;s\u00a0<a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1548\">full post<\/a>. \u00a0We will see how Steenrod&#8217;s Theorem follows&#8230; next time.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Last time, I explained some of the strange things that happen with projective limits of topological spaces: they can be empty, even if all the spaces in the given projective system are non-empty and all bonding maps are surjective, and &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1571\">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":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1571","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\/1571","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=1571"}],"version-history":[{"count":3,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/1571\/revisions"}],"predecessor-version":[{"id":1574,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/1571\/revisions\/1574"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1571"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1571"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1571"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}