Priestley space

Template:Short description 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. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality ("Priestley duality") between the category of Priestley spaces and the category of bounded distributive lattices.

Definition

A Priestley space is an ordered topological space Template:Math, i.e. a set Template:Math equipped with a partial order Template:Math and a topology Template:Math, satisfying the following two conditions:

1. Template:Math is compact.
2. If $x\le ̸y$, then there exists a clopen up-set Template:Math of Template:Math such that Template:Math and Template:Math. (This condition is known as the Priestley separation axiom.)

Properties of Priestley spaces

• Each Priestley space is Hausdorff. Indeed, given two points Template:Math of a Priestley space Template:Math, if Template:Math, then as Template:Math is a partial order, either $x\le ̸y$ or $y\le ̸x$. Assuming, without loss of generality, that $x\le ̸y$, (ii) provides a clopen up-set Template:Math of Template:Math such that Template:Math and Template:Math. Therefore, Template:Math and Template:Math are disjoint open subsets of Template:Math separating Template:Math and Template:Math.
• Each Priestley space is also zero-dimensional; that is, each open neighborhood Template:Math of a point Template:Math of a Priestley space Template:Math contains a clopen neighborhood Template:Math of Template:Math. To see this, one proceeds as follows. For each Template:Math, either $x\le ̸y$ or $y\le ̸x$. By the Priestley separation axiom, there exists a clopen up-set or a clopen down-set containing Template:Math and missing Template:Math. The intersection of these clopen neighborhoods of Template:Math does not meet Template:Math. Therefore, as Template:Math is compact, there exists a finite intersection of these clopen neighborhoods of Template:Math missing Template:Math. This finite intersection is the desired clopen neighborhood Template:Math of Template:Math contained in Template:Math.

It follows that for each Priestley space Template:Math, the topological space Template:Math 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 Template:Math be a Priestley space.

(a) For each closed subset Template:Math of Template:Math, both Template:Math and Template:Math are closed subsets of Template:Math.

(b) Each open up-set of Template:Math is a union of clopen up-sets of Template:Math and each open down-set of Template:Math is a union of clopen down-sets of Template:Math.

(c) Each closed up-set of Template:Math is an intersection of clopen up-sets of Template:Math and each closed down-set of Template:Math is an intersection of clopen down-sets of Template:Math.

(d) Clopen up-sets and clopen down-sets of Template:Math form a subbasis for Template:Math.

(e) For each pair of closed subsets Template:Math and Template:Math of Template:Math, if Template:Math, then there exists a clopen up-set Template:Math such that Template:Math and Template:Math.

A Priestley morphism from a Priestley space Template:Math to another Priestley space Template:Math is a map Template:Math which is continuous and order-preserving.

Let Pries denote the category of Priestley spaces and Priestley morphisms.

Connection with spectral spaces

Priestley spaces are closely related to spectral spaces. For a Priestley space Template:Math, let Template:Math denote the collection of all open up-sets of Template:Math. Similarly, let Template:Math denote the collection of all open down-sets of Template:Math.

Theorem: If Template:Math is a Priestley space, then both Template:Math and Template:Math are spectral spaces.

Conversely, given a spectral space Template:Math, let Template:Math denote the patch topology on Template:Math; that is, the topology generated by the subbasis consisting of compact open subsets of Template:Math and their complements. Let also Template:Math denote the specialization order of Template:Math.

Theorem: If Template:Math is a spectral space, then Template:Math 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.

Connection with bitopological spaces

Priestley spaces are also closely related to bitopological spaces.

Theorem: If Template:Math is a Priestley space, then Template:Math is a pairwise Stone space. Conversely, if Template:Math is a pairwise Stone space, then Template:Math is a Priestley space, where Template:Math is the join of Template:Math and Template:Math and Template:Math is the specialization order of Template:Math.

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:

$𝐒𝐩𝐞𝐜\cong 𝐏𝐫𝐢𝐞𝐬\cong 𝐏𝐒𝐭𝐨𝐧𝐞$

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.

See also

• Spectral space
• Pairwise Stone space
• Distributive lattice
• Stone duality
• Duality theory for distributive lattices

References

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