{"id":3615,"date":"2014-01-16T21:47:27","date_gmt":"2014-01-16T21:47:27","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3615"},"modified":"2014-01-16T21:49:10","modified_gmt":"2014-01-16T21:49:10","slug":"limits-of-quasi-compact-spaces","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3615","title":{"rendered":"Limits of quasi-compact spaces"},"content":{"rendered":"<p>The limit of a directed inverse system of quasi-compact spaces need not be quasi-compact. Danger Will Robinson!<\/p>\n<p>Nice exercise: what happens with an inverse limit of spectral spaces with spectral maps? A <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/08YF\">spectral space<\/a> is a topological space which is sober, has a basis of quasi-compact opens, and is such that the intersection of any two quasi-compact opens is quasi-compact; actually Hochster showed these are always homeomorphic to spectra of rings.<\/p>\n<p>As usual: don&#8217;t answer if you know the answer&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The limit of a directed inverse system of quasi-compact spaces need not be quasi-compact. Danger Will Robinson! Nice exercise: what happens with an inverse limit of spectral spaces with spectral maps? A spectral space is a topological space which is &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3615\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-3615","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3615","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3615"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3615\/revisions"}],"predecessor-version":[{"id":3620,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3615\/revisions\/3620"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3615"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3615"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3615"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}