University of Birmingham > Talks@bham > Lab Lunch > Revisiting the relation between subspaces and sublocales

Revisiting the relation between subspaces and sublocales

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If you have a question about this talk, please contact Todd Waugh Ambridge.

Pointfree topology regards certain order theoretical structures called locales as being abstract topological spaces. This is done in virtue of an adjunction between the category Top of topological spaces and Loc of locales, where the left adjoint sends each space to its locale O(X) of open sets. Locales of this form are called spatial.

One of the most significant features distinguishing pointfree topology from classical point-set one is a mismatch between the subspaces of a topological space X, and the sublocales of the corresponding locale O(X). A locale may have abstract sublocales which do not have any point-set analogue, namely the nonspatial sublocales. One striking consequence of this is that 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: how do we characterize those spaces X such that the collection of sublocales of O(X) is a perfect representation of the subspaces of 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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