---
tag: ZGBV8V3823
kind: concept
name: Priestley space
msc: [06D50, 54F05, "06E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Priestley_space
same_as:
  wikidata: Q7242916
  wikipedia: Priestley space
---

## Summary

A Priestley space is a compact topological space with a partial order in which any two points $x \not\le y$ are separated by a clopen upper set.
Priestley spaces are the duals of bounded [[distributive lattices|0HM0HQCK6N]] in [[Priestley duality|SA22SEC75K]].
Forgetting the order, every Priestley space is a [[Stone space|ZV38HA5PAQ]], and a Stone space with the equality order is a Priestley space.
So Priestley spaces extend Stone duality from Boolean algebras to distributive lattices.

## Explanations

### L1 human

A Priestley space carries two structures at once: a compact, very disconnected topology and an order.
One rule ties them together: whenever $x$ is not below $y$, some set that is open, closed and closed upward contains $x$ but not $y$.
A finite poset with the discrete topology is the simplest example.
The clopen upper sets of a Priestley space form a distributive lattice, and every bounded distributive lattice arises in this way.