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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