Variants of normality and their duals: a pointfree unification of insertion and extension theorems for real-valued functions

Resumen

Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore be studied in parallel. In this talk we discuss the source of this ‘duality’ and show that each pair of parallel results can be framed by the ‘same’ proof. The key tools for this purpose are relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixedclass of complemented sublocales of the given locale) that bring and extend to locale theory avariety of well-known classical variants of normality and upper and lower semicontinuities in ailluminating unified manner. This approach allows us to unify under a single localic proof a great variety ofclassical insertion results, as well as their corresponding extension results.