Higher order dissolutions and Boolean coreflections of locales

If a locale has a Boolean coreflection, then this coreflection can be constructed by iterating the dissolution functor Amaps toAd until it stabilizes at a Boolean locale. In this paper we characterize α-soluble locales, i.e. locales whose αth dissolution is Boolean, for α≤4. There are examples of spaces whose chain of dissolutions stabilizes at the first, second, or third dissolution. Beyond that no examples are known. We also give a sufficient condition for insolubility which implies that the locale of rational numbers has no Boolean coreflection, and hence that the only metrizable spaces with a Boolean coreflection are the scattered ones.