Let $X$ be a compact Hausdorff space and $C(X)$ the algebra of continuous functions $X \longrightarrow \mathbb{C}$ . Recall that $C(X)$ is a vector lattice with the usual order: $f\leq g \Longleftrightarrow g-f$ takes positive (or zero) values.

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Theorem - If every subset of $C(X)$ that is bounded from above has a least upper bound (i.e. $C(X)$ is a bounded complete lattice), then $X$ is extremally disconnected.