nLab Priestley space

Definition

A Priestley space is a compact topological space with a partial order $\leq$ which satisfies the Priestley separation axiom:

• If $x \nleq y$, then there exists a clopen upper set $U$ of $X$ such that $x \in U$ and $y \notin U$.

Properties

Every Priestley space is a sober topological space. Furthermore, every Priestley space is a Hausdorff space and a Stone space.

Examples

• Every finite poset is a Priestley space.

Relation to coherent spaces

Theorem

Every Priestley space is a coherent space in two different ways: where the open set topology are given by the upper sets or the lower sets respectively.

Theorem

Every coherent space is a Priestley space whose topology is given by the topology generated by the subbasis? consisting of compact open subsets and their complements and whose partial order is given by the specialization order.

Category of Priestley spaces

The category of Priestley spaces is the category whose objects are Priestley spaces and whose morphisms are monotonic continuous functions between Priestley spaces.

The category of Priestley spaces is the opposite category of DistLat, the category of distributive lattices. This phenomenon is called Priestley duality, and is a Stone-type duality akin to the one between Stone spaces and Boolean algebras or locales and frames. The category of Priestley spaces is thus equivalent to the category of coherent spaces.

Related concepts

• distributive lattice

• coherent space

References

• Wikipedia, Priestley space

• Andrej Bauer, Karin Cvetko-Vah, Mai Gehrke, Sam van Gool, Ganna Kudryavtseva, A non-commutative Priestley duality, Topology and its Applications, Volume 160, Issue 12, 1 August 2013, Pages 1423-1438 (doi:10.1016/j.topol.2013.05.012, arXiv:1206.5848)

• William H Cornish, Peter R Fowler, Coproducts of de morgan algebras, Bulletin of the Australian Mathematical Society, 16(01):1–13, 1977. (pdf)

• William H Cornish, Peter R Fowler, Coproducts of kleene algebras, Journal of the Australian Mathematical Society (Series A), 27(02):209–220, 1979 (pdf)