Esakia space

In mathematics, Esakia spaces are special ordered topological spaces introduced and studied by Leo Esakia in 1974. Esakia spaces play a fundamental role in the study of Heyting algebras, primarily by virtue of the Esakia duality—the dual equivalence between the category of Heyting algebras and the category of Esakia spaces.

Definition
For a partially ordered set $(X, ≤)$ and for $x ∈ X$, let $↓x = {y ∈ X : y≤ x}$ and let $↑x = {y ∈ X : x≤ y}$. Also, for $A⊆ X$, let $↓A = {y ∈ X : y ≤ x for some x ∈ A}$ and $↑A = {y ∈ X : y≥ x for some x ∈ A}$.

An Esakia space is a Priestley space $(X,τ,≤)$ such that for each clopen subset $C$ of the topological space $(X,τ)$, the set $↓C$ is also clopen.

Equivalent definitions
There are several equivalent ways to define Esakia spaces.

Theorem: Given that $(X,τ)$ is a Stone space, the following conditions are equivalent:


 * (i) $(X,τ,≤)$ is an Esakia space.


 * (ii) $↑x$ is closed for each $x ∈ X$ and $↓C$ is clopen for each clopen $C⊆ X$.


 * (iii) $↓x$ is closed for each $x ∈ X$ and $↑cl(A) = cl(↑A)$ for each $A⊆ X$ (where $cl$ denotes the closure in $X$).


 * (iv) $↓x$ is closed for each $x ∈ X$, the least closed set containing an up-set is an up-set, and the least up-set containing a closed set is closed.

Since Priestley spaces can be described in terms of spectral spaces, the Esakia property can be expressed in spectral space terminology as follows: The Priestley space corresponding to a spectral space $X$ is an Esakia space if and only if the closure of every constructible subset of $X$ is constructible.

Esakia morphisms
Let $(X,≤)$ and $(Y,≤)$ be partially ordered sets and let $f : X → Y$ be an order-preserving map. The map $f$ is a bounded morphism (also known as p-morphism) if for each $x ∈ X$ and $y ∈ Y$, if $f(x)≤ y$, then there exists $z ∈ X$ such that $x≤ z$ and $f(z) = y$.

Theorem: The following conditions are equivalent:


 * (1) $f$ is a bounded morphism.


 * (2) $f(↑x) = ↑f(x)$ for each $x ∈ X$.


 * (3) $f^{&minus;1}(↓y) = ↓f^{&minus;1}(y)$ for each $y ∈ Y$.

Let $(X, τ, ≤)$ and $(Y, τ', ≤)$ be Esakia spaces and let $f : X → Y$ be a map. The map $f$ is called an Esakia morphism if $f$ is a continuous bounded morphism.