{"id":3873,"date":"2021-05-20T10:49:12","date_gmt":"2021-05-20T08:49:12","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=3873"},"modified":"2023-03-20T11:27:58","modified_gmt":"2023-03-20T10:27:58","slug":"the-sorgenfrey-line-is-not-consonant","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=3873","title":{"rendered":"The Sorgenfrey line is not consonant"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In Exercise 5.4.12 of the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>, I ask the reader to prove that neither the space of rationals, <strong>Q<\/strong>, nor the Sorgenfrey line, <strong>R<\/strong><sub>\u2113<\/sub>, is consonant.  But the proofs I had in mind were much too simple-minded to stand any chance of succeeding, hence I classified this as important blooper #5 in the list of <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=12\">errata<\/a>.  Good news: Showing that <strong>R<\/strong><sub>\u2113<\/sub> is not consonant is not that hard, finally.  I will explain the argument in the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=3685\" data-type=\"page\" data-id=\"3685\">full post<\/a>.  This will also be an excuse to explain some additional topological properties of <strong>R<\/strong><sub>\u2113<\/sub>, an introduction to hereditary Lindel\u00f6fness (we will see that <strong>R<\/strong><sub>\u2113<\/sub> is hereditarily Lindel\u00f6f, although it is not second-countable), and a few additional things in the appendices.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Exercise 5.4.12 of the book, I ask the reader to prove that neither the space of rationals, Q, nor the Sorgenfrey line, R\u2113, is consonant. But the proofs I had in mind were much too simple-minded to stand any &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=3873\">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":[11,18,16,43],"class_list":["post-3873","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-consonance","tag-counterexample","tag-powerdomain","tag-valuation"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/3873","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=3873"}],"version-history":[{"count":5,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/3873\/revisions"}],"predecessor-version":[{"id":5887,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/3873\/revisions\/5887"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3873"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3873"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3873"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}