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:

(i) is an Esakia space.

(ii) is closed for each and is clopen for each clopen .

(iii) is closed for each and for each (where denotes the closure in ).

(iv) is closed for each , 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 is an Esakia space if and only if the closure of every constructible subset of is constructible.

Esakia morphisms

Let and be partially ordered sets and let be an order-preserving map. The map is a bounded morphism (also known as p-morphism) if for each and , if , then there exists such that and .

Theorem: The following conditions are equivalent:

(1) is a bounded morphism.

(2) for each .

(3) for each .

Let and be Esakia spaces and let be a map. The map is called an Esakia morphism if is a continuous bounded morphism.

Notes

References

• Esakia, L. (1974). Topological Kripke models. Soviet Math. Dokl., 15 147–151.
• Esakia, L. (1985). Heyting Algebras I. Duality Theory (Russian). Metsniereba, Tbilisi.