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A bitopological point-free approach to compactifications

Klinke, Olaf Karl (2012)
Ph.D. thesis, University of Birmingham.

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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.

Type of Work:Ph.D. thesis.
Supervisor(s):Jung , Achim and Escardo, Martin and Vickers, Steven
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Computer Science
Subjects:GA Mathematical geography. Cartography
QA75 Electronic computers. Computer science
QA76 Computer software
Institution:University of Birmingham
ID Code:3470
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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