Could not include topos theory - contents
A locale is topological, or spatial, if it is isomorphic to for some topological space .
A locale has enough points if, given any two opens and in , if (hence iff) precisely the same points of belong to as belong to .
The following conditions are all logically equivalent on a locale :
(It would be nice to state this as a theorem and put in a proof.)
Basically, what is going on here is that we have an idempotent adjunction from topological spaces to locales, and the topological locales comprise the image of this adjunction. The corresponding condition on topological spaces is being sober.
The terms ‘topological locale’ and ‘spatial locale’ can be confusing; they suggest a locale in Top or in some category Sp of spaces, which is not correct. Instead, the adjective ‘topological’ and ‘spatial’ should be taken in the same vein as ‘localic’ in ‘localic topos’ or ‘topological’ in ‘topological convergence’. These two terms also suggest that the study of other locales is not part of topology or that these other locales are not spaces, which is also incorrect.
The really clear term for a topological locale is ‘locale with enough points to separate the opens’, but ‘locale with enough points’ should be unambiguous. However, it is still a bit long. The shortest term, ‘spatial locale’, is probably also the most common. Occasionally one sees ‘spacial’ instead of ‘spatial’, but this might just be a misspelling.