Esakia space
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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 (X, ≤) and for x∈ X, let ↓x = {y∈ X : y≤ x} and let ↑x = {y∈ X : x≤ y}. Also, for A⊆ X, let ↓A = {y∈ X : y ≤ x for some x∈ A} and ↑A = {y∈ X : y≥ x for some x∈ A}.
An Esakia space is a Priestley space (X,τ,≤) such that for each clopen subset C of the topological space (X,τ), the set ↓C is also clopen.
Equivalent definitions
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,≤) is an Esakia space.
- (ii) ↑x is closed for each x∈ X and ↓C is clopen for each clopen C⊆ X.
- (iii) ↓x is closed for each x∈ X and ↑cl(A) = cl(↑A) for each A⊆ X (where cl denotes the closure in X).
- (iv) ↓x is closed for each x∈ 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]