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We give geometric characterisations of patch and Lawson topologies in thecontext of predicative point-free topology using the constructive notion oflocated subset. We present the patch topology of a stably locally compactformal topology by a geometric theory whose models are the points of the giventopology that are located, and the Lawson topology of a continuous lattice by ageometric theory whose models are the located subsets of the given lattice. Wealso give a predicative presentation of the frame of perfect nuclei on a stablylocally compact formal topology, and show that it is essentially the same asour geometric presentation of the patch topology. Moreover, the construction ofLawson topologies naturally induces a monad on the category of compact regularformal topologies, which is shown to be isomorphic to the Vietoris monad.

Geometric theories of patch and Lawson topologies