{"id":2728,"date":"2020-08-23T11:00:03","date_gmt":"2020-08-23T09:00:03","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2728"},"modified":"2023-03-20T11:32:48","modified_gmt":"2023-03-20T10:32:48","slug":"chains-and-nested-spaces","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2728","title":{"rendered":"Chains and nested spaces"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">A chain is a totally ordered poset, and a <em>nested space<\/em> is a topological space whose lattice of open sets is a chain.  That may seem like a curious notion, although you might say that the Scott topology on the real line makes it a nested space\u2014so you know that there at least one natural example of the concept.  I will show that nested spaces and chains have very strong topological properties.  To start with, I will show you why every chain is a <em>continuous<\/em> poset.  I will then tell you how nested spaces arise from the study of so-called <em>minimal<\/em>\u00a0T<sub>0\u00a0<\/sub>and T<sub><em>D<\/em>\u00a0<\/sub>topologies, as first explored by R. E. Larson in 1969.  And I will conclude with a simple proof of a recent theorem by Mike Mislove.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2702\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A chain is a totally ordered poset, and a nested space is a topological space whose lattice of open sets is a chain. That may seem like a curious notion, although you might say that the Scott topology on the &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2728\">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":[51,21,52,53,43],"class_list":["post-2728","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-chain","tag-continuous-lattice","tag-minimal-topology","tag-td-space","tag-valuation"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2728","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=2728"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2728\/revisions"}],"predecessor-version":[{"id":2729,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2728\/revisions\/2729"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2728"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2728"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2728"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}