= Solution
The <Priestley dual space> $\widehat L$ of a <distributive lattice> $L$ has all <prime filter of a distributive lattice>[prime filters] of $L$ as its points. Its order is inclusion. For each $a\in L$, put
$$
a^*=\{P\in\widehat L:a\in P\}.
$$
The topology is generated by the sets $a^*$ and their complements. Each $a^*$ is therefore a <clopen up-set>, and these sets separate points and order. With this topology and order, $\widehat L$ is a compact totally order-disconnected ordered space.
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