{"id":2217,"date":"2020-01-20T18:10:40","date_gmt":"2020-01-20T17:10:40","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2217"},"modified":"2023-03-20T11:36:28","modified_gmt":"2023-03-20T10:36:28","slug":"borel-sets-analytic-sets-and-the-baire-property","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2217","title":{"rendered":"Borel sets, analytic sets, and the Baire property"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Let me venture into the realm of \u03c3-algebras.  Yes, you might say, that is measure theory, not topology&#8230; but topology plays an important role in measure theory and, for that matter, descriptive set theory.  I will tell you about sets with <em>the Baire property<\/em>.  Those are pretty simple objects, or at least they appear to be simpler than Borel sets, but we will see that this is the other way around: <em>all<\/em> Borel sets have the property of Baire.  The proof is pretty easy, as well.  I will also spend some more time to explain a more complicated result, due to O.M. Nikod\u00fdm, and which says that all A-sets have the property of Baire as well, in a second-countable space.  None of that ever uses any Hausdorffness, or in fact any separation property whatsoever.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2185\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let me venture into the realm of \u03c3-algebras. Yes, you might say, that is measure theory, not topology&#8230; but topology plays an important role in measure theory and, for that matter, descriptive set theory. I will tell you about sets &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2217\">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":[64,62,63],"class_list":["post-2217","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-analytic-set","tag-baire-property","tag-borel-set"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2217","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=2217"}],"version-history":[{"count":3,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2217\/revisions"}],"predecessor-version":[{"id":2222,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2217\/revisions\/2222"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2217"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2217"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2217"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}