In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Esakia spaces are special ordered topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s introduced and studied by Leo Esakia in 1974. Esakia spaces play a fundamental role in the study of Heyting algebra

Heyting algebra

In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...

s, primarily by virtue of the Esakia duality

Esakia duality

In mathematics, Esakia duality is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heyting algebras via Esakia spaces....

 --- the dual equivalence

Equivalence of categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...

 between the category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 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

Priestley space

In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributive lattices...

  such that for each clopen

Clopen set

In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...

 subset of the topological space , the set is also clopen.

Equivalent definitions

There are several equivalent ways to define Esakia spaces.

Theorem: The following conditions are equivalent:
is an Esakia space.
is closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 for each and is clopen for each clopen .
is closed for each and for each (where denotes the closure

Closure operator

In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....

 in ).
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.

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:
is a bounded morphism.
for each .
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.