---
tag: 996BWM6ZQ7
kind: concept
name: Esakia space
msc: ["06D20", "06D50", "03G10"]
specializes: [ZGBV8V3823]
sources:
  - url: https://en.wikipedia.org/wiki/Esakia_space
same_as:
  wikidata: Q5396698
  wikipedia: Esakia space
---

## Summary

An Esakia space is a [[Priestley space|ZGBV8V3823]] in which the down-set of every clopen set is again clopen.
Esakia spaces are the spaces dual to [[Heyting algebras|NG943XNQWH]], by [[Esakia duality|CD9Y7YP4EJ]].
The Heyting algebra of an Esakia space is its algebra of clopen up-sets.
They connect the algebraic semantics of [[intuitionistic logic|Q11N8X99FK]] with its [[Kripke semantics|NGHWZ9Y7ZV]].

## Explanations

### L1 human

An Esakia space is a compact space whose points are also partially ordered, in a way that fits the topology well.
One can think of the points as states of knowledge, where moving up in the order means learning more.
The extra rule says that the set of states below a clopen set is again clopen.
This rule makes the implication of two clopen up-sets again a clopen up-set.
Every finite partially ordered set, with the discrete topology, is an Esakia space.