October 1997 (Series of two talks)
Anneliese Schauerte , University of Cape Town


A biframe is simultaneously a generalization of a frame (locale) and a bitopological space. To be specific, a biframe L is a triple L=(L0,L1,L2) where L1 and L2 are subframes of the frame L0, which together generate it. A typical example (derived from a bispace) might have L0 consisting of all open sets of the reals, and L1 and L2 consisting of intervals of the form (-infinity,b) and (a,infinity), respectively. Functions between biframes are morphisms in the category BiFrm if they are frame homomorphisms, and preserve first and second parts..
In some areas the theory of biframes closely parallels that of frames and topological spaces; in others it is remarkably different. We will give examples of both.

Talk 1: Compactifications of biframes
All compactifications of a biframe can be represented by means of strong inclusions: these are pairs of relations given on the biframe itself, requiring no information about any other biframe. In this setting, zero-dimensional compactifications are easy to identify; smallest compactifications less so.

Talk 2: Coherent and continuous biframes

There is a category equivalence between coherent biframes and distributive bilattices, and an analogous one between supercoherent biframes and bisemilattices. These lead to definitions of continuity and stable continuity, and characterization in terms of projectivity properties.

Booleanization in biframes seems to have rather different problems and possibilities to that in frames. Criteria such as "unique dense Boolean quotient of L" do not apply; and various conditions defining weakly open maps that are equivalent in frames are not so, in the biframe setting.