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The frame generated by closed sublocales

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If you have a question about this talk, please contact Paul Taylor.

The original motivation came from the following phenomenon:

• a frame is subfit iff each of its open sublocales is a join of closed ones

• a frame is fit iff each of its closed sublocales is a meet of open ones.

Now the latter is equivalent with the formally stronger statement

• a frame is fit iff each of its sublocales is a meet of open ones.

The question is what happens if we assume that each sublocale whatsoever is a join of closed ones. It turned out that it characterized the scattered frames, a point-free counterpart of the standard concept of scattered spaces.

While proving this it turned out that the system S c (L) of joins of closed sublocales is always a frame (embedded into the co-frame of all sublocales S(L)). If the original frame L is subfit, it is, moreover, also a co-frame, a quotient (sub-colocale) of S(L), indeed a Boolean algebra, and L is naturally embedded into it (by sending an element to the associated open sublocale).

Thus we have a Boolean extension of a subfit frame. It has turned out that

• it can be used with advantage in modelling “non-continuous frame ho- momorphisms”,

• it is (isomorphic) with the Booleanization of the frame of nuclei, the canonicity of which was pointed out by Johnstone (and thus eluminat- ing its nature, see also the following point)

• and it is the maximal essential extension of L;

• furthermore, if X is a T 1 -space, the S c (Ω(L)) picks out precisely the sublocales induced by subspaces from the coframe S(Ω(L)) of all sublo- cales of Ω(L).

This talk is part of the Theoretical computer science seminar series.

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