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.
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.
There are several equivalent ways to define Esakia spaces.
Theorem:[2] 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.[3]
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:[4] 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.