{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T19:12:33Z","timestamp":1725390753467},"reference-count":0,"publisher":"EasyChair","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>Several familiar results about normal and extremally disconnected (classical or pointfree) spaces<\/jats:p><jats:p>shape the idea that the two notions are somehow dual to each other and can therefore be studied in<\/jats:p><jats:p>parallel. In this talk we discuss the source of this \u2018duality\u2019 and show that each pair of parallel<\/jats:p><jats:p>results can be framed by the \u2018same\u2019 proof. The key tools for this purpose are relative notions<\/jats:p><jats:p>of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed<\/jats:p><jats:p>class of complemented sublocales of the given locale) that bring and extend to locale theory a<\/jats:p><jats:p>variety of well-known classical variants of normality and upper and lower semicontinuities in a<\/jats:p><jats:p>illuminating unified manner. This approach allows us to unify under a single localic proof a great variety of<\/jats:p><jats:p>classical insertion results, as well as their corresponding extension results.<\/jats:p>","DOI":"10.29007\/4lmb","type":"proceedings-article","created":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T22:59:06Z","timestamp":1516748346000},"page":"171-165","source":"Crossref","is-referenced-by-count":0,"title":["Variants of normality and their duals: a pointfree unification of insertion and extension theorems for real-valued functions"],"prefix":"10.29007","volume":"25","author":[{"given":"Jorge","family":"Picado","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Javier","family":"Guti\u00e9rrez Garc\u00eda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"11545","event":{"name":"TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic"},"container-title":["EPiC Series in Computing"],"original-title":[],"deposited":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T23:01:21Z","timestamp":1516748481000},"score":1,"resource":{"primary":{"URL":"https:\/\/easychair.org\/publications\/paper\/4Wn"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[null]]},"references-count":0,"URL":"https:\/\/doi.org\/10.29007\/4lmb","relation":{},"ISSN":["2398-7340"],"issn-type":[{"type":"print","value":"2398-7340"}],"subject":[]}}