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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.

This construction yields a Priestley space: a compact ordered topological space in which suitable clopen upward-closed sets separate distinct comparable-order failures.

A Priestley space is a compact topological partially ordered set that is totally order-disconnected: if $x\nleqq y$, a clopen upward-closed set contains $x$ and omits $y$. A Priestley dual space is a space of this kind constructed from the prime filters of a bounded distributive lattice.