{"id":782,"date":"2015-10-25T17:26:45","date_gmt":"2015-10-25T16:26:45","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=782"},"modified":"2017-10-30T17:09:12","modified_gmt":"2017-10-30T16:09:12","slug":"ideal-domains-i","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=782","title":{"rendered":"Ideal domains I"},"content":{"rendered":"<p>A few months ago, <a title=\"Keye Martin\" href=\"https:\/\/www.keyemartin.com\/\">Keye Martin<\/a> drew my attention to his results on so-called ideal models of spaces [1]. \u00a0Ideal domains are incredibly specific dcpos: they are defined as dcpos where each non-finite element is maximal. \u00a0Despite this, Keye Martin was able to show that: (1) every space that has an \u03c9-continuous model has an ideal model, that is, a model that is an ideal domain; (2) the metrizable spaces that have an ideal model are exactly the completely metrizable spaces.<\/p>\n<p>I will try to expose a few of his ideas here. \u00a0I will probably betray him a lot. \u00a0For example, I will not talk about measurements (one of Keye&#8217;s inventions), and I will not stress the role of Choquet-completeness to go beyond &#8220;Lawson at the top&#8221; domains, or the role of <em>G<\/em><sub>\u03b4<\/sub> subsets so much.<\/p>\n<p>Last minute update: I had also tried to extend whatever I could to the case of quasi-metric, not just metric, spaces, but I did not manage to do so. Read the (corrected)\u00a0<a title=\"Ideal models I\" href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A few months ago, Keye Martin drew my attention to his results on so-called ideal models of spaces [1]. \u00a0Ideal domains are incredibly specific dcpos: they are defined as dcpos where each non-finite element is maximal. \u00a0Despite this, Keye Martin &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=782\">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-782","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\/782","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=782"}],"version-history":[{"count":4,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/782\/revisions"}],"predecessor-version":[{"id":797,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/782\/revisions\/797"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=782"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=782"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=782"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}