{"id":1281,"date":"2017-10-30T17:03:35","date_gmt":"2017-10-30T16:03:35","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1281"},"modified":"2017-10-30T17:07:35","modified_gmt":"2017-10-30T16:07:35","slug":"a-characterization-of-fac-spaces","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1281","title":{"rendered":"A characterization of FAC spaces"},"content":{"rendered":"<p>In the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1147\">open problem<\/a> section, I defined a <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1141\">FAC space<\/a> as a topological space in which every closed subspace is a finite union of irreducible closed subspaces. FAC is for &#8220;finite antichain property&#8221;, since it generalizes the following theorem, due to Erd\u0151s and Tarski (1943): a poset has the finite antichain property (namely, all its antichains are finite) if and only if its downwards-closed subsets are finite unions of ideals. I asked about a similar characterization of FAC <em>spaces<\/em>. Let me give a positive answer to that in the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1272\">full post<\/a>!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the open problem section, I defined a FAC space as a topological space in which every closed subspace is a finite union of irreducible closed subspaces. FAC is for &#8220;finite antichain property&#8221;, since it generalizes the following theorem, due &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=1281\">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-1281","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\/1281","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=1281"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/1281\/revisions"}],"predecessor-version":[{"id":1282,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/1281\/revisions\/1282"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1281"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1281"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1281"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}