Compactly generated Hausdorff locales

AbstractWe say that a Hausdorff locale is compactly generated if it is the colimit of the diagram of its compact sublocales connected by inclusions. We show that this is the case if and only if the natural map of its frame of opens into the second Lawson dual is an isomorphism. More generally, for any Hausdorff locale, the second dual of the frame of opens gives the frame of opens of the colimit. In order to arrive at this conclusion, we generalize the Hofmann–Mislove–Johnstone theorem and some results regarding the patch construction for stably locally compact locales