{"id":573,"date":"2024-03-10T18:36:43","date_gmt":"2024-03-10T18:36:43","guid":{"rendered":"https:\/\/pri.tum.ac.ke\/?page_id=573"},"modified":"2024-03-10T18:36:44","modified_gmt":"2024-03-10T18:36:44","slug":"on-%ce%b3-compactifiable-locales","status":"publish","type":"page","link":"https:\/\/pri.tum.ac.ke\/?page_id=573","title":{"rendered":"On \u03b3-Compactifiable Locales"},"content":{"rendered":"\n<p>A. FWARU, T. DUBE AND M. MUNYWOKI<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Abstract<\/h3>\n\n\n\n<p>Let <em>L <\/em>be a completely regular locale. <em>L <\/em>is called countably com- pactifiable (abbreviated <em>\u03b3<\/em><em>-compactifiable<\/em>) if there exists a countably compact completely regular locale <em>M <\/em>such that <em>L<\/em>is a dense sublocale of <em>M<\/em>, and every countably compact closed sublocale of <em>L<\/em>is closed in <em>M<\/em>. This class of locales properly encompasses the class of locales of open sets of Tychonoff spaces with analogous features introduced by K.Morita (1973) in Countably-compactifiable spaces. Morita established that if well-separated spaces <em>X<\/em>and <em>Y<\/em>with count- ably compactifications <em>R<\/em>and <em>S<\/em>, respectively, and the product space <em>R<\/em>\u00d7 <em>S <\/em>is countably compactifiable, then the product space <em>X<\/em>\u00d7 <em>Y<\/em>inherits this property. Our aim is to extend Morita\u2019s findings to locales and frames.<\/p>\n\n\n\n<p>Let <em>\u03b2L<\/em>denote the Stone-C<sup>\u02c7<\/sup>ech compactification of <em>L<\/em>. We explore various<\/p>\n\n\n\n<p>operations that preserve the countably compactifiable property, we show that If a locale <em>L <\/em>is <em>\u03b3<\/em>-compactifiable, then there is a <em>\u03b3<\/em>-compactification <em>S <\/em>of <em>L <\/em>such that <em>L <\/em>\u2286 <em>S <\/em>\u2286 <em>\u03b2L <\/em>. Additionally, we address a fundamental question: Can the product of two countably compact (or countably compactifiable) locales be <em>\u03b3<\/em>-compactifiable ?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A. FWARU, T. DUBE AND M. MUNYWOKI Abstract Let L be a completely regular locale. L is called countably com- pactifiable (abbreviated \u03b3-compactifiable) if there [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":113,"menu_order":7,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-573","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/573","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=573"}],"version-history":[{"count":1,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/573\/revisions"}],"predecessor-version":[{"id":574,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/573\/revisions\/574"}],"up":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/113"}],"wp:attachment":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}