{"id":6599,"date":"2023-03-20T11:14:18","date_gmt":"2023-03-20T10:14:18","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6599"},"modified":"2023-03-20T11:16:16","modified_gmt":"2023-03-20T10:16:16","slug":"localic-products-and-till-plewes-game","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6599","title":{"rendered":"Localic products and Till Plewe&#8217;s game"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Products in the category of locales resemble, but do not coincide with products in the category of topological spaces.  Till Plewe has a nice explanation to this, as I will explain in this month&#8217;s post: the localic product of two topological spaces coincides with their topological products if and only if player II has a winning strategy in a certain game, which I have already described last month.  As a consequence, we will obtain that the localic product of <em>S<\/em><sub>0<\/sub> with itself is not its topological product (a result due to Matthew de Brecht), we will retrieve that the localic product of <strong>Q<\/strong> and of <strong>R<\/strong>\u2013<strong>Q<\/strong> differs from their topological product (and more generally, a result of John Isbell&#8217;s), and finally that the localic product of <strong>Q<\/strong> with itself differs from the topological product, and that <strong>Q<\/strong> is not consonant&#8230; with a much, much simpler proof than those I have ever mentioned here.  The idea of that argument is due to Matthew de Brecht.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6400\" data-type=\"page\" data-id=\"6400\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Products in the category of locales resemble, but do not coincide with products in the category of topological spaces. Till Plewe has a nice explanation to this, as I will explain in this month&#8217;s post: the localic product of two &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6599\">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":[11,18,41,10,42,38],"class_list":["post-6599","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-consonance","tag-counterexample","tag-frame","tag-game","tag-locale","tag-ordinal"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6599","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=6599"}],"version-history":[{"count":6,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6599\/revisions"}],"predecessor-version":[{"id":6605,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6599\/revisions\/6605"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6599"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6599"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6599"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}