Results of Bunge and Funk and of Johnstone, providing constructively sound descriptions of the global points of the lower and upper powerlocales, are extended here to describe the generalized points and proved in a way that displays in a symmetric fashion two complementary treatments of frames: as suplattices and as preframes. Also described here are the points of the Vietoris powerlocale.

In each of two special cases, an exponential $D ($ being the Sierpiński locale) is shown to be homeomorphic to a powerlocale: to the lower powerlocale when D is discrete, and to the upper powerlocale when D is compact regular.