{"id":574,"date":"2015-02-23T16:28:21","date_gmt":"2015-02-23T15:28:21","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=574"},"modified":"2022-11-19T15:30:49","modified_gmt":"2022-11-19T14:30:49","slug":"iwamuras-lemma-kowalskys-theorem-and-ordinals","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=574","title":{"rendered":"Iwamura&#8217;s Lemma, Markowsky&#8217;s Theorem, and ordinals"},"content":{"rendered":"<p>On p.61 of the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>, there is a remark that the dcpos are exactly the chain-complete posets.\u00a0 This is a theorem by George Markowsky (1976).\u00a0 It is time I explained seriously how this worked.\u00a0 The first step is Iwamura&#8217;s Lemma (1944), which states that every directed subset decomposes as the union of a small chain of small directed subsets.<\/p>\n<p>The reason I did not put the proof of that result in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a> is because it rests on using ordinals, and I did not want to introduce ordinals, specially if they served for only one result.\u00a0 I&#8217;m attacking the problem head-front in the <a title=\"Iwamura\u2019s Lemma, Kowalsky\u2019s Theorem and ordinals\" href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=563\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>On p.61 of the book, there is a remark that the dcpos are exactly the chain-complete posets.\u00a0 This is a theorem by George Markowsky (1976).\u00a0 It is time I explained seriously how this worked.\u00a0 The first step is Iwamura&#8217;s Lemma &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=574\">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-574","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\/574","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=574"}],"version-history":[{"count":4,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/574\/revisions"}],"predecessor-version":[{"id":5960,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/574\/revisions\/5960"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=574"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=574"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=574"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}