  The Vietoris space of a Stone space plays an important role in the
coalgebraic approach to modal logic. When generalizing this to positive modal
logic, there is a variety of relevant hyperspace constructions based on various
topologies on a Priestley space and mechanisms to topologize the hyperspace of
closed sets. A number of authors considered hyperspaces of Priestley spaces and
their application to the coalgebraic approach to positive modal logic. A
mixture of techniques from category theory, pointfree topology, and Priestley
duality have been employed. Our aim is to provide a unifying approach to this
area of research relying only on a basic familiarity with Priestley duality and
related free constructions of distributive lattices.
