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Description: |
One of the main features distinguishing pointfree topology from classical point-set one is that in the pointfree setting a space (i.e. a locale) may have abstract subspaces (sublocales) which do not have any point-set analogue. The lattice of sublocales of a locale will not in general be a powerset, but a coframe. It is also particularly striking that in pointfree topology every locale has a smallest dense sublocale, a result which is far from true in the point-set setting, where intersections of dense subspaces need not be dense.
The relation between point-set subspaces and pointfree ones has already been explored in the literature; one of the main questions has been: what are those spatial locales O(X) such that their coframe of sublocales S(O(X)) is the same as the powerset P(X)?
We introduce a new method to answer this and related questions, based on considering sobrifications of subspaces on one side, and spatializations of sublocales on the other.
By using this method, we obtain new proofs of characterization theorems that link P(X) and S(O(X)). In particular we are able to characterize in several ways, without appealing to point-set reasoning, those spatial locales O(X) such that the collection of spatial sublocales is the same as the powerset P(X), as well as those such that the collection of all sublocales coincides with P(X).
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Date: |
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| Start Time: |
15:00 |
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Speaker: |
Anna Laura Suarez (Univ. Birmingham, UK)
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Institution: |
PhD student at University of Birmingham
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Place: |
Sala 5.5
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| Research Groups: |
-Algebra, Logic and Topology
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See more:
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<Main>
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