1 June 2011

16:00

to

17:30

Olaf Klinke

Abstract

<p>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.
</p>
<p>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: </p>
<ul>
<li>The real line with the bitopology of upper and lower semicontinuity
admits precisely one compactification, the extended reals.</li>
<li>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.</li></ul>