[1] Priestley spaces play a fundamental role in the study of distributive lattices.
For a Priestley space (X,τ,≤), let τu denote the collection of all open up-sets of X.
Similarly, let τd denote the collection of all open down-sets of X. Theorem:[5] If (X,τ,≤) is a Priestley space, then both (X,τu) and (X,τd) are spectral spaces.
Conversely, if (X,τ1,τ2) is a pairwise Stone space, then (X,τ,≤) is a Priestley space, where τ is the join of τ1 and τ2 and ≤ is the specialization order of (X,τ1).
One of the main consequences of the duality theory for distributive lattices is that each of these categories is dually equivalent to the category of bounded distributive lattices.