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

5 pages•Published: July 28, 2014

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. 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 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 a great variety of
classical insertion results, as well as their corresponding extension results.

Keyphrases: completely separated sublocales, continuous extension, continuous real function, extremally disconnected frame, frame and locale, katetov relation, normal continuous real function, normal frame, pointfree topology, regular continuous real function, semicontinuous real function, sublocale lattice, zero continuous real function

In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 171-175.

Links:	https://easychair.org/publications/paper/4Wn
	https://doi.org/10.29007/4lmb