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 and for, let and let . Also, for, let and.
An Esakia space is a Priestley space such that for each clopen subset of the topological space, the set is also clopen.

Equivalent definitions

There are several equivalent ways to define Esakia spaces.
Theorem: Given that is a Stone space, the following conditions are equivalent:

Esakia morphisms

Let and be partially ordered sets and let be an order-preserving map. The map is a bounded morphism if for each and, if, then there exists such that and.
Theorem: The following conditions are equivalent:
Let and be Esakia spaces and let be a map. The map is called an Esakia morphism if is a continuous bounded morphism.