Priestley duality

Priestley duality is a dual equivalence between bounded distributive lattices and compact totally order-disconnected ordered topological spaces.

The Priestley dual space $\hat{L}$ of a bounded distributive lattice $L$ consists of its prime filters, ordered by inclusion and topologized by the sets $a^{\ast }=\{P:a\in P\}$ and their complements.

A clopen up-set is both clopen in the topology and upward closed in the order. The clopen up-sets of a Priestley space form a distributive lattice under intersection and union.

The Stone map sends an element $a$ of a distributive lattice to the clopen up-set $a^{\ast }=\{P:a\in P\}$ of its Priestley dual space. The Stone prime filter theorem makes this lattice homomorphism injective.