Esakia space

In mathematics, Esakia spaces are special ordered topological spaces introduced and studied by Leo Esakia in 1974.^[1] 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[edit]

For a partially ordered set $(X, \leq)$ and for $x \in X$ , let $↓ x = {y \in X : y \leq x$ } and let $↑ x = {y \in X : x \leq y$ }. Also, for $A \subseteq X$ , let $↓ A = {y \in X : y \leq x for some x \in A$ } and $↑ A = {y \in X : y \geq x for some x \in A$ }.

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

Equivalent definitions[edit]

There are several equivalent ways to define Esakia spaces.

Theorem:^[2] Given that $(X, τ)$ is a Stone space, the following conditions are equivalent:

(i)

(X, τ,\leq)

is an Esakia space.

(ii)

↑ x

is closed for each

x \in X

and

↓ C

is clopen for each clopen

C \subseteq X

.

(iii)

↓ x

is closed for each

x \in X

and

↑cl(A) = cl(↑ A)

for each

A \subseteq X

(where

cl

denotes the closure in

X

).

(iv)

↓ x

is closed for each

x \in 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.^[3]

Esakia morphisms[edit]

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

Theorem:^[4] The following conditions are equivalent:

(1)

f

is a bounded morphism.

(2)

f(↑ x) = ↑f(x)

for each

x \in X

.

(3)

f -1 (↓ y) = ↓f -1 (y)

for each

y \in Y

.

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

Notes[edit]

^ Esakia (1974)
^ Esakia (1974), Esakia (1985).
^ see section 8.3 of Dickmann, Schwartz, Tressl (2019)
^ Esakia (1974), Esakia (1985).

References[edit]

Esakia, L. (1974). Topological Kripke models. Soviet Math. Dokl., 15 147–151.
Esakia, L. (1985). Heyting Algebras I. Duality Theory (Russian). Metsniereba, Tbilisi.
Dickmann, Max; Schwartz, Niels; Tressl, Marcus (2019). Spectral Spaces. New Mathematical Monographs. Vol. 35. Cambridge: Cambridge University Press. doi:10.1017/9781316543870. ISBN 9781107146723.

[1] Esakia (1974)

[2] Esakia (1974), Esakia (1985).

[3] see section 8.3 of Dickmann, Schwartz, Tressl (2019)

[4] Esakia (1974), Esakia (1985).

[1]

[2]

[3]

[4]