In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them.[1] Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality ("Priestley duality"[2]) between the category of Priestley spaces and the category of bounded distributive lattices.[3] [4]
A Priestley space is an ordered topological space, i.e. a set equipped with a partial order and a topology, satisfying the following two conditions:
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\scriptstyley\not\lex
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\scriptstyley\not\lex
It follows that for each Priestley space, the topological space is a Stone space; that is, it is a compact Hausdorff zero-dimensional space.
Some further useful properties of Priestley spaces are listed below.
Let be a Priestley space.
(a) For each closed subset of, both and are closed subsets of .
(b) Each open up-set of is a union of clopen up-sets of and each open down-set of is a union of clopen down-sets of .
(c) Each closed up-set of is an intersection of clopen up-sets of and each closed down-set of is an intersection of clopen down-sets of .
(d) Clopen up-sets and clopen down-sets of form a subbasis for .
(e) For each pair of closed subsets and of, if, then there exists a clopen up-set such that and .
A Priestley morphism from a Priestley space to another Priestley space is a map which is continuous and order-preserving.
Let Pries denote the category of Priestley spaces and Priestley morphisms.
Priestley spaces are closely related to spectral spaces. For a Priestley space, let denote the collection of all open up-sets of . Similarly, let denote the collection of all open down-sets of .
Theorem:[5] If is a Priestley space, then both and are spectral spaces.
Conversely, given a spectral space, let denote the patch topology on ; that is, the topology generated by the subbasis consisting of compact open subsets of and their complements. Let also denote the specialization order of .
Theorem:[6] If is a spectral space, then is a Priestley space.
In fact, this correspondence between Priestley spaces and spectral spaces is functorial and yields an isomorphism between Pries and the category Spec of spectral spaces and spectral maps.
Priestley spaces are also closely related to bitopological spaces.
Theorem:[7] If is a Priestley space, then is a pairwise Stone space. Conversely, if is a pairwise Stone space, then is a Priestley space, where is the join of and and is the specialization order of .
The correspondence between Priestley spaces and pairwise Stone spaces is functorial and yields an isomorphism between the category Pries of Priestley spaces and Priestley morphisms and the category PStone of pairwise Stone spaces and bi-continuous maps.
Thus, one has the following isomorphisms of categories:
Spec\congPries\congPStone
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.