Editing Heyting algebra (section)

== Topological representation and duality theory==
Every Heyting algebra {{math|''H''}} is naturally isomorphic to a bounded sublattice {{math|''L''}} of open sets of a topological space {{math|''X''}}, where the implication <math>U\to V</math> of {{math|''L''}} is given by the interior of <math>(X\setminus U)\cup V</math>. 
More precisely, {{math|''X''}} is the [[spectral space]] of prime [[ideal (order theory)|ideals]] of the bounded lattice {{math|''H''}} and {{math|''L''}} is the lattice of open and quasi-compact subsets of {{math|''X''}}.

More generally, the category of Heyting algebras 
is dually equivalent to the category of Heyting spaces.<ref>see section 8.3 in * {{cite book | last1=Dickmann | first1=Max | last2=Schwartz | first2= Niels | last3=Tressl | first3= Marcus | title=Spectral Spaces| doi=10.1017/9781316543870 | year=2019 | publisher=[[Cambridge University Press]] | series=New Mathematical Monographs | volume=35 | location=Cambridge | isbn=9781107146723 | s2cid=201542298 }} </ref> 
This duality can be seen as restriction of the classical [[Stone duality]] of bounded distributive lattices to the (non-full) subcategory of Heyting algebras.

Alternatively,  the category of Heyting algebras is dually equivalent to the category of [[Esakia space]]s. This is called [[Esakia duality]].