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).
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