Our study of topological spaces presented as convergence structures shaped the idea that "topological spaces are generalised orders", and eventually revealed the following analogies. down-closed subset = filter of opens non-empty down-closed subset = proper filter of opens directed down-closed subset = prime filter of opens upper bound = limit point supremum = smallest limit point cocomplete ordered set = continuous lattice directed cocomplete ordered set = stably compact space completely distributive lattice = ??? continuous directed cocomplete ordered set = ??? In this seminar we try to remove the two remaining question marks. Furthermore, we show that the category of distributive spaces is dually equivalent to a category of frames by simply observing that both sides represent the idempotent splitting completion of the same category.
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