{"id":5439,"date":"2022-05-20T11:26:49","date_gmt":"2022-05-20T09:26:49","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=5439"},"modified":"2023-02-20T19:29:55","modified_gmt":"2023-02-20T18:29:55","slug":"compact-scattered-subsets-and-a-topological-game","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=5439","title":{"rendered":"Compact scattered subsets and a topological game"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Showing that <strong>Q<\/strong> is not consonant is quite an ordeal.  I have finally managed to understand one of the existing proofs of this fact, due to Costantini and Watson.  This would be a bit too long to cover entirely in one post, so the bulk of the explanation will be for another time.  Instead, I will explain why the compact subsets of\u00a0<strong>Q<\/strong> are all\u00a0<em>scattered<\/em>, and what it means, but the important point of this month&#8217;s post is that, reading between the lines, the Costantini-Watson argument relies on a property that I will characterize through the use of a topological game\u00a0<em>G<\/em>(<em>K<\/em>), resembling the strong Choquet game, in which we will see that player I has a winning strategy if\u00a0<em>K<\/em>\u00a0is compact and scattered\u2014and that is an if <em>and only if<\/em> in any T<sub>2<\/sub> space.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4926\" data-type=\"page\" data-id=\"4926\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Showing that Q is not consonant is quite an ordeal. I have finally managed to understand one of the existing proofs of this fact, due to Costantini and Watson. This would be a bit too long to cover entirely in &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=5439\">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":[25,10,38,26],"class_list":["post-5439","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-compact","tag-game","tag-ordinal","tag-scattered"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/5439","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=5439"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/5439\/revisions"}],"predecessor-version":[{"id":5440,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/5439\/revisions\/5440"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5439"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5439"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5439"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}