{"id":2698,"date":"2020-07-20T18:33:33","date_gmt":"2020-07-20T16:33:33","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2698"},"modified":"2023-03-20T11:31:59","modified_gmt":"2023-03-20T10:31:59","slug":"td-spaces","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2698","title":{"rendered":"TD spaces"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In any topological space, the closure of any one-element set {<em>x<\/em>} is also its downward closure \u2193<em>x<\/em> with respect to the specialization preordering. A T<sub><em>D<\/em><\/sub> space is a topological space in which, for every point <em>x<\/em>, \u2193<em>x<\/em> \u2013 {<em>x<\/em>} is closed, too.  This seemingly weird concept was introduced by Aull and Thron in a 1962 paper, but it has funny and interesting applications, notably in the comparison of the notions of subspaces and of sublocales, and in Thron&#8217;s so-called lattice equivalence problem.  I will also mention the Skula topology again&#8230; Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2626\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In any topological space, the closure of any one-element set {x} is also its downward closure \u2193x with respect to the specialization preordering. A TD space is a topological space in which, for every point x, \u2193x \u2013 {x} is &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2698\">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":[53],"class_list":["post-2698","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-td-space"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2698","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=2698"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2698\/revisions"}],"predecessor-version":[{"id":2699,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2698\/revisions\/2699"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2698"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2698"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2698"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}