On the relation between subspaces and sublocales (Q1979313)

In this investigation, the author shows the instrisic relation between sublocales and subspaces. This situation leads to the question: for which locales (frames) all sublocales (quotients) are spatial? In order to give a full answer, the author introduces the set of all spatial sublocales as a sublocale of the coframe of all sublocales. At the end, it is shown that for a spatial frame its point space is scattered exactly when it is \(T_{D}\), and this holds exactly when all the primes of the frame are completely prime. The investigation shows how localic-methods and frame-theoretic methods interplay in a such good way.