A bitopological point-free approach to compactifications

Abstract

We study structures called d-frames which were developed by the last two authors for a bitopological treatment of Stone duality. These structures consist of a pair of frames thought of as the opens of two topologies, together with two relations which serve as abstractions of disjointness and covering of the space. With these relations, the topological separation axioms regularity and normality have natural analogues in d-frames. We develop a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames. Given that normality of topological spaces does not behave well with respect to products and subspaces, probably the most surprising result is this: The category of d-frames has a normal coreflection, and the Stone–Čech compactification factors through it. Moreover, any compactification can be obtained by first producing a regular normal d-frame and then applying the Stone–Čech compactification to it. Our bitopological compactification subsumes all classical compactifications of frames as well as Smythʼs stable compactification.