Forthcoming Seminars › Seminar Series

Analytic Topology in Mathematics and Computer Science

Wed, 04/05/2011
16:00 		Dr Secil Tokgoz (Turkey) Analytic Topology in Mathematics and Computer Science Add to calendar L3
Some results on submaximal spaces
A topological space $ (X,\tau) $ is submaximal if $ \tau $ is the maximal element of $ [{\tau}_{s}] $. Submaximality was first defined and characterized by Bourbaki. Since then, some mathematicians presented several characterizations of submaximal spaces. In this paper, we will attempt to develop the concept of submaximality and offer some new results. Furthermore, some results concerning $ \alpha $-scattered space will be obtained.
Wed, 01/06/2011
16:00 		Olaf Klinke (University of Birmingham) Analytic Topology in Mathematics and Computer Science Add to calendar L3
A bitopological point-free approach to compactification
It is known for long that the set of possible compactifications of a topological space (up to homeomorphism) is in order-preserving bijection to "strong inclusion" relations on the lattice of open sets. Since these relations do not refer to points explicitly, this bijection has been generalised to point-free topology (a.k.a. locales). The strong inclusion relations involved are typically "witnessed" relations. For example, the Stone-Cech compactification has a strong inclusion witnessed by real-valued functions. This makes it natural to think of compactification in terms of d-frames, a category invented by Jung and Moshier for bitopological Stone duality. Here, a witnessed strong inclusion is inherent to every object and plays a central role. We present natural analogues of the topological concepts regularity, normality, complete regularity and compactness in d-frames. Compactification is then a coreflection into the category of d-frames dually equivalent to compact Hausdorff spaces. The category of d-frames has a few surprising features. Among them are:
  • The real line with the bitopology of upper and lower semicontinuity admits precisely one compactification, the extended reals.
  • Unlike in the category of topological spaces (or locales), there is a coreflection into the subcategory of normal d-frames, and every compactification can be factored as "normalisation" followed by Stone-Cech compactification.

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