Seminário de Álgebra, Lógica e Topologia

Cozero elements of a frame play an important role in point-free topology. The set of cozero elements, Coz L, of a frame L is a sub σ-frame of L (that is, a sublattice closed under countable suprema and finite infima). Moreover, the lattice Coz L join-generates the frame L if and only if L is completely regular (a result analogous to the classical one for completely regular topological spaces).
Within the setting of completely regular frames, we consider the Dedekind-MacNeille completion of the lattice of cozero elements, which in this context coincides with the Bruns-Lakser construction. The frame obtained through this process turns out to be a sublocale of the original frame, specifically, the smallest sublocale containing Coz L. A central aim of the talk is to compare this sublocale with the original frame and to present results and examples showing when the two coincide and when they differ.
Viewing Coz L both as a join-generating sublattice and through its completion leads naturally to two related but distinct classes of frames: cozero frames and perfectly regular frames. We will examine the latter carefully and provide illustrative examples, not only in frames but how this reflects in the classical topological context.
This is an ongoing joint work with Guram Bezhanishvili and Joanne Walters-Wayland.