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CATEGORIES:Lab Lunch
SUMMARY:Localic completion of generalized metric spaces II
: Powerlocales - Steve Vickers
DTSTART:20090915T120000Z
DTEND:20090915T130000Z
UID:TALK69AT
URL:/talk/index/69
DESCRIPTION:That's the paper that I have just had accepted\, a
nd you can find it on\nmy web page. However\, I wo
n't get far explaining it without also also\nexpla
ining its predecessor "Localic completion of gener
alized metric\nspaces I". In fact I'll probably sp
end most of my time explaining what\nthe titles me
an.\n\n"Localic completion"\, roughly speaking\, m
eans describing the points of\nthe completion of a
metric space as the models of a logical theory -
but\nthe logic here is the so-called geometric log
ic. The logical theory then\nhas the virtue of des
cribing both the points and the topology all in on
e\ngo. Such localic techniques have been used in d
enotational semantics\nsince the 1980s and are par
ticularly useful in constructive reasoning.\n\nFor
the metric spaces (and these can be understood in
a highly\ngeneralized sense that includes asymmet
ric metrics) the points of the\ncompletion come ou
t as "Cauchy filters of formal balls". Perhaps I'l
l\nhave time to illustrate this with the reals as
completion of the rationals.\n\nThe powerlocale pa
per uses this localic approach for hyperspaces\, i
.e.\nspaces whose points are subspaces of some oth
er space. Specifically\, if\nX is the metric space
then the hyperspaces of its completion can be\nde
scribed by completing the finite powerset of X\, w
ith appropriate\nmetrics. Interestingly\, the corr
esponding subspaces of the completion do\nnot have
to be finite.\n\nI will also say something about
the 10 years it took me to get\nthe paper publishe
d.\n\n*(Sandwiches will be provided.)*
LOCATION:CS 124
CONTACT:Dan Ghica
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