A bitopological point-free approach to compactifications
Olaf Klinke, School of Computer Science, University of Birmingham
Date and time: Friday 17th September 2010 at 12:30
Location: 124, School of Computer Science
Host: topology lab lunch
We study structures called 'd-frames' which were developed by Jung and Moshier 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 normal topological spaces do 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. Moreover, any compactification can be obtained by first producing a regular normal d-frame and then applying the Stone-Cech compactification to it. Our bitopological compactification subsumes all classical compactifications of frames as well as Smyth's stable compactifications.