Priestley dual space
= Priestley dual space
{c}
The Priestley dual space $\widehat L$ of a bounded <distributive lattice> $L$ consists of its <prime filter of a distributive lattice>[prime filters], ordered by inclusion and topologized by the sets $a^*=\{P:a\in P\}$ and their complements.
This construction yields a <Priestley space>: a compact ordered topological space in which suitable clopen upward-closed sets separate distinct comparable-order failures.