# Priestley space

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]

## Definition

A *Priestley space* is an *ordered topological space* (*X*,*τ*,≤), i.e. a set *X* equipped with a partial order ≤ and a topology *τ*, satisfying
the following two conditions:

## Properties of Priestley spaces

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

It follows that for each Priestley space (*X*,*τ*,≤), the topological space (*X*,*τ*) 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 (*X*,*τ*,≤) be a Priestley space.

- (a) For each closed subset
*F*of*X*, both ↑*F*= {*x*∈*X*:*y*≤*x*for some*y*∈*F*} and ↓*F*= {*x*∈*X*:*x*≤*y*for some*y*∈*F*} are closed subsets of*X*.

- (a) For each closed subset

- (b) Each open up-set of
*X*is a union of clopen up-sets of*X*and each open down-set of*X*is a union of clopen down-sets of*X*.

- (b) Each open up-set of

- (c) Each closed up-set of
*X*is an intersection of clopen up-sets of*X*and each closed down-set of*X*is an intersection of clopen down-sets of*X*.

- (c) Each closed up-set of

- (d) Clopen up-sets and clopen down-sets of
*X*form a subbasis for (*X*,*τ*).

- (d) Clopen up-sets and clopen down-sets of

- (e) For each pair of closed subsets
*F*and*G*of*X*, if ↑*F*∩ ↓*G*= ∅, then there exists a clopen up-set*U*such that*F*⊆*U*and*U*∩*G*= ∅.

- (e) For each pair of closed subsets

A **Priestley morphism** from a Priestley space (*X*,*τ*,≤) to another Priestley space (*X*′,*τ*′,≤′) is a map f : *X* → *X*′ 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 (*X*,*τ*,≤), let *τ*^{u} denote the collection of all open up-sets of *X*. Similarly, let *τ*^{d} denote the collection of all open down-sets of *X*.

**Theorem:**[5]
If (*X*,*τ*,≤) is a Priestley space, then both (*X*,*τ*^{u}) and (*X*,*τ*^{d}) are spectral spaces.

Conversely, given a spectral space (*X*,*τ*), let *τ*^{#} denote the patch topology on *X*; that is, the topology generated by the subbasis consisting of compact open subsets of (*X*,*τ*) and their complements. Let also ≤ denote the specialization order of (*X*,*τ*).

**Theorem:**[6]
If (*X*,*τ*) is a spectral space, then (*X*,*τ*^{#},≤) 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:**[7]
If (*X*,*τ*,≤) is a Priestley space, then (*X*,*τ*^{u},*τ*^{d}) is a pairwise Stone space. Conversely, if (*X*,*τ*_{1},*τ*_{2}) is a pairwise Stone space, then (*X*,*τ*,≤) is a Priestley space, where *τ* is the join of *τ*_{1} and *τ*_{2} and ≤ is the specialization order of (*X*,*τ*_{1}).

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:

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

## Notes

- Priestley, (1970).
- Cignoli, R.; Lafalce, S.; Petrovich, A. (September 1991). "Remarks on Priestley duality for distributive lattices".
*Order*.**8**(3): 299–315. doi:10.1007/BF00383451. - Cornish, (1975).
- Bezhanishvili et al. (2010)
- Cornish, (1975). Bezhanishvili et al. (2010).
- Cornish, (1975). Bezhanishvili et al. (2010).
- Bezhanishvili et al. (2010).

## References

- Priestley, H. A. (1970). "Representation of distributive lattices by means of ordered Stone spaces".
*Bull. London Math. Soc*.**2**(2): 186–190. doi:10.1112/blms/2.2.186. - Priestley, H. A. (1972). "Ordered topological spaces and the representation of distributive lattices" (PDF).
*Proc. London Math. Soc*.**24**(3): 507–530. doi:10.1112/plms/s3-24.3.507. hdl:10338.dmlcz/134149. - Cornish, W. H. (1975). "On H. Priestley's dual of the category of bounded distributive lattices".
*Mat. Vesnik*.**12**(27): 329–332. - Hochster, M. (1969). "Prime ideal structure in commutative rings".
*Trans. Amer. Math. Soc*.**142**: 43–60. doi:10.1090/S0002-9947-1969-0251026-X. - Bezhanishvili, G.; Bezhanishvili, N.; Gabelaia, D.; Kurz, A (2010). "Bitopological duality for distributive lattices and Heyting algebras" (PDF).
*Mathematical Structures in Computer Science*.**20**. - Dickmann, Max; Schwartz, Niels; Tressl, Marcus (2019).
*Spectral Spaces*. New Mathematical Monographs.**35**. Cambridge: Cambridge University Press. doi:10.1017/9781316543870. ISBN 978-1-107-14672-3.