---
tag: 59PG49KH6V
kind: definition
name: Esakia space
msc: ["06D20", "06D50", "03G10"]
defines: 996BWM6ZQ7
uses: [ZGBV8V3823, DQ4E55T3BW, 0QV2JS9RJ3, ETCE287E2R]
sources:
  - url: https://en.wikipedia.org/wiki/Esakia_space
---

## Definition

Let $(X, \tau, \le)$ be a [[Priestley space|ZGBV8V3823]].
So $(X, \tau)$ is a compact topological space and $\le$ is a [[partial order|0QV2JS9RJ3]] on $X$.
Also, for all $x, y \in X$ with $x \not\le y$, there is a [[clopen|DQ4E55T3BW]] [[up-set|ETCE287E2R]] $U$ with $x \in U$ and $y \notin U$.
For $S \subseteq X$, write ${\downarrow} S = \set{x \in X : x \le s \text{ for some } s \in S}$.
The space $(X, \tau, \le)$ is an *Esakia space* if ${\downarrow} C$ is clopen for every clopen set $C \subseteq X$.

## Remarks

In a Priestley space, ${\downarrow} C$ is always closed when $C$ is closed; the extra condition is that it is also open when $C$ is clopen.
For clopen up-sets $U$ and $V$ of an Esakia space, the set $X \setminus {\downarrow}(U \setminus V)$ is a clopen up-set, and it is the Heyting implication $U \to V$.

Example: a finite poset $(P, \le)$ with the discrete topology.
Every subset is clopen, so the separation condition holds with $U = {\uparrow} x$, and every down-set is clopen.

Non-example: let $X = \set{a_1, a_2, \ldots} \cup \set{\infty, b}$, where $a_n \to \infty$ as in the one-point compactification of $\N$, and $b$ is an isolated point.
Order $X$ by $\infty \le b$ and no other strict relations.
This is a Priestley space, but $\set{b}$ is clopen and ${\downarrow}\set{b} = \set{\infty, b}$ is not open, since every neighbourhood of $\infty$ contains some $a_n$.