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CATEGORIES:Lab Lunch
SUMMARY:Revisiting the relation between subspaces and subl
ocales - Anna Laura Suarez (University of Birmingh
am)
DTSTART:20200123T120000Z
DTEND:20200123T130000Z
UID:TALK4121AT
URL:/talk/index/4121
DESCRIPTION:Pointfree topology regards certain order theoretic
al structures called locales as being abstract top
ological spaces. This is done in virtue of an adju
nction between the category Top of topological spa
ces and Loc of locales\, where the left adjoint se
nds each space to its locale O(X) of open sets. Lo
cales of this form are called spatial.\n\nOne of t
he most significant features distinguishing pointf
ree topology from classical point-set one is a mis
match between the subspaces of a topological space
X\, and the sublocales of the corresponding local
e O(X). A locale may have abstract sublocales whic
h do not have any point-set analogue\, namely the
nonspatial sublocales. One striking consequence of
this is that every locale has a smallest dense su
blocale\, a result which is far from true in the p
oint-set setting\, where intersections of dense su
bspaces need not be dense. \n\nThe relation betwee
n point-set subspaces and pointfree ones has alrea
dy been explored in the literature. One of the mai
n questions has been: how do we characterize those
spaces X such that the collection of sublocales o
f O(X) is a perfect representation of the subspace
s of X?\n\nWe introduce a new method to answer thi
s and related questions\, based on considering sob
rifications of subspaces on one side\, and spatial
izations of sublocales on the other.\nBy using thi
s method\, we obtain new proofs of characterizatio
n theorems that link P(X) and S(O(X)). In particul
ar we are able to characterize in several ways\, w
ithout appealing to point-set reasoning\, those sp
atial locales O(X) such that the collection of spa
tial sublocales is the same as the powerset P(X)\,
as well as those such that the collection of all
sublocales coincides with P(X).
LOCATION:CS 217
CONTACT:Todd Waugh Ambridge
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