Abstrakt
We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given object B is an element of B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely thefact that a compact metric space is never the Cech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., tint a compact met rizable locale is never the (cell-Stone compactification of a proper sublocale.
This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice principles, every compact regular locale is the topology of a compact Ilausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces.
We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character.
At the opposite extreme from these results, we show that an extremally disconnected locale is a compacHlication of each of its dense sublocales. Finally, we analyse the same phenomena, also in the pointiree setting, for the O-dimensional compact reflection arid for the Lindelof reflection.