A bitopological point-free approach to compactifications

Abstract

This thesis extends the concept of compactifications of topological spaces to a setting where spaces carry a partial order and maps are order-preserving. The main tool is a Stone-type duality between the category of d-frames, which was developed by Jung and Moshier, and bitopological spaces. We demonstrate that the same concept that underlies d-frames can be used to do recover short proofs of well-known facts in domain theory. In particular we treat the upper, lower and double powerdomain constructions in this way. The classification of order-preserving compactifications follows ideas of B. Banaschewski and M. Smyth. Unlike in the categories of spaces or locales, the lattice-theoretic notion of normality plays a central role in this work. It is shown that every compactification factors as a normalisation followed by the maximal compactification, the Stone-Cech compactification. Sample applications are the Fell compactification and a stably compact extension of algebraic domains.