Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 120 / 3 / b / Solution

The Priestley dual space $\hat{L}$ of a distributive lattice $L$ has all prime filters of $L$ as its points. Its order is inclusion. For each $a\in L$, putThe topology is generated by the sets $a^{\ast }$ and their complements. Each $a^{\ast }$ is therefore a clopen up-set, and these sets separate points and order. With this topology and order, $\hat{L}$ is a compact totally order-disconnected ordered space.