On the bitopological nature of Stone Duality

Based on the theory of frames we introduce a Stone duality for bitopological spaces. The central concept is that of a d-frame, which axiomatises the two open set lattices. Exploring the resulting concept of d-sobriety we find this to be a much more inclusive concept than usual sobriety. Spatial d-frames suggest additional axioms that lead us to define reasonable d-frames; these have an alternative presentation as partial frames. We explore natural notions of regularity and compactness for bitopological spaces, and their manifestation in d-frames. This yields the machinery to locate precisely within this general landscape a number of classical Stonetype dualities, namely, those of Stone for Boolean algebras and bounded distributive lattices, those of the present authors for strong proximity lattices (with negation), and the duality of classical frames. The general duality can be given a logical reading by viewing the open sets of one topology as positive extents of formulas, and those of the other topology as negative extents. This point of view emphasises the fact that formulas may be undecidable in certain states and may be self-contradictory in others. We also obtain two natural orders on the set of formulas, one related to Scott’s information order and the other being the usual logical implication. The interplay between the two can be said to be the main organising principle of this study. 1