{"id":5785,"date":"2022-10-19T16:35:31","date_gmt":"2022-10-19T14:35:31","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=5785"},"modified":"2023-02-20T19:19:16","modified_gmt":"2023-02-20T18:19:16","slug":"strongly-compact-sets-and-the-double-hyperspace-construction","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=5785","title":{"rendered":"Strongly compact sets and the double hyperspace construction"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The notion of strongly compact set is due to Reinhold Heckmann.  A few months ago, I said that I would explain why the sobrification of the space <strong>Q<\/strong><sub>fin<\/sub>(<em>X<\/em>) of finitary compact sets on a sober space <em>X<\/em> is not the Smyth hyperspace <strong>Q<\/strong>(<em>X<\/em>), rather its subspace of strongly compact saturated sets <strong>Q<\/strong><sub>s<\/sub>(<em>X<\/em>).  This what I will start with.  I will then present a funny other case where strongly compact sets are required.  There is a long line of research purporting to show that, for certain spaces <em>X<\/em>, the Smyth and Hoare hyperspace constructions commute, namely that <strong>QH<\/strong><em>X<\/em> and <strong>HQ<\/strong><em>X<\/em> are homeomorphic.  The most complete such result is due to Matthew de Brecht and Tatsuji Kawai in 2019; they showed that this is the case <em>exactly<\/em> when <em>X<\/em> is consonant.  I will give a simplified exposition of their proof, and I will show that essentially the same proof shows that <strong>Q<\/strong><sub>s<\/sub><strong>H<\/strong><em>X<\/em> and <strong>HQ<\/strong><sub>s<\/sub><em>X<\/em> are homeomorphic, for <em>every<\/em> topological space whatsoever.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=5736\" data-type=\"page\" data-id=\"5736\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The notion of strongly compact set is due to Reinhold Heckmann. A few months ago, I said that I would explain why the sobrification of the space Qfin(X) of finitary compact sets on a sober space X is not the &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=5785\">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":[17,15,16],"class_list":["post-5785","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-compactness","tag-hyperspace","tag-powerdomain"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/5785","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=5785"}],"version-history":[{"count":3,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/5785\/revisions"}],"predecessor-version":[{"id":5788,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/5785\/revisions\/5788"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5785"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5785"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5785"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}