{"id":4664,"date":"2022-01-20T19:47:58","date_gmt":"2022-01-20T18:47:58","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=4664"},"modified":"2023-02-20T19:28:21","modified_gmt":"2023-02-20T18:28:21","slug":"irredundant-families-the-smyth-powerdomain-the-lyu-jia-theorem-and-the-baby-groemer-theorem","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=4664","title":{"rendered":"Irredundant families, the Smyth powerdomain, the Lyu-Jia theorem, and the baby Groemer theorem"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">A \u2229-semilattice of sets is a family of sets that is closed under finite intersections, and it is irredundant if and only if <em>all<\/em> its non-empty elements are irreducible.  That sounds like a ridiculously overconstrained notion, but I will give two applications of the notion.  One, which I will actually present last, is a baby version of the so-called Groemer theorem.  This baby Groemer theorem is non-trivial, but has an amazingly simple proof, due to Klaus Keimel; we have used it to prove non-Hausdorff generalizations of a line of theorems due to Choquet, Kendall and Matheron.  The other application is due to Zhenchao Lyu and Xiaodong Jia.  The Smyth powerdomain <strong>Q<\/strong>(<em>X<\/em>) of a space <em>X<\/em> is locally compact if and only if <em>X<\/em> is, and they were interested in knowing whether the same would happen with &#8220;core-compact&#8221; instead of &#8220;locally compact&#8221;.  The answer is no, and this rests on a clever use of irredundancy, and the existence of a core-compact, non-locally compact space.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4610\" data-type=\"page\" data-id=\"4610\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A \u2229-semilattice of sets is a family of sets that is closed under finite intersections, and it is irredundant if and only if all its non-empty elements are irreducible. That sounds like a ridiculously overconstrained notion, but I will give &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=4664\">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":[17,30,15,16],"class_list":["post-4664","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-compactness","tag-core-compactness","tag-hyperspace","tag-powerdomain"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/4664","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=4664"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/4664\/revisions"}],"predecessor-version":[{"id":4665,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/4664\/revisions\/4665"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4664"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4664"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4664"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}