A bitopological point-free approach to compactification
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Wed, 01/06/2011 16:00 |
Olaf Klinke (University of Birmingham) |
Analytic Topology in Mathematics and Computer Science  |
L3 |
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:
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