On the parallel between normality and extremal disconnectedness

Abstract

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. This paper investigates the source of this 'duality' and shows 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 fixed class of complemented sublocales of the given locale) that bring and extend to locale theory a variety of well-known classical variants of normality and upper and lower semicontinuities in a illuminating unified manner. This approach allows us to unify under a single localic proof all classical insertion, as well as their corresponding extension results.

Perfect locales and localic real functions

“…The results above have interesting counterparts in the dual case of extremally disconnected locales (in the vein of the approach in [10] and [8]).…”

“…The reader will have no difficulties then in obtaining the dual of Theorem 3 (since the notions of upper A-semicontinuity and lower A-semicontinuity are dual to each other [10]): Corollary 4. Let L be a locale and let A be a sublattice of L. Then A is countably complemented and extremally disconnected in L if and only if for any lower A-semicontinuous u ∈ F(L) and any upper A-semicontinuous l ∈ F(L) such that u ≤ l, there exists an A-continuous f ∈ F(L) such that u ≤ f ≤ l and ι(u, f ) = ι(f, l) = ι(u, l).…”

“…The paper [10] proposes a unified treatment of several variants of semicontinuities and continuities and their insertion results, viz. A-continuity of an f ∈ F(L) where A is a class of complemented sublocales of L. We end this paper with some notes about the application of this approach to the results above.…”