The Stacks project

Remark 5.26.10. Let $X$ be a quasi-compact Hausdorff space. Let $\kappa$ be an infinite cardinal bigger or equal than the cardinality of $X$. Then the cardinality of the minimal quasi-compact, Hausdorff, extremally disconnected cover $X' \to X$ (Lemma 5.26.9) is at most $2^{2^\kappa }$. Namely, choose a subset $S \subset X'$ mapping bijectively to $X$. By minimality of $X'$ the set $S$ is dense in $X'$. Thus $|X'| \leq 2^{2^\kappa }$ by Lemma 5.25.1.