The Stacks project

Example 5.26.8. We can use Proposition 5.26.6 to see that the Stone-Čech compactification $\beta (X)$ of a discrete space $X$ is extremally disconnected. Namely, let $f : Y \to \beta (X)$ be a continuous surjection where $Y$ is quasi-compact and Hausdorff. Then we can lift the map $X \to \beta (X)$ to a continuous (!) map $X \to Y$ as $X$ is discrete. By the universal property of the Stone-Čech compactification we see that we obtain a factorization $X \to \beta (X) \to Y$. Since $\beta (X) \to Y \to \beta (X)$ equals the identity on the dense subset $X$ we conclude that we get a section. In particular, we conclude that the Stone-Čech compactification of a discrete space is totally disconnected, whence profinite (see discussion following Definition 5.26.1 and Lemma 5.22.2).