Corollary 34.6.5.label Let $(X, \cm, \mu)$ be a localisable measure space, then $\Omega(L^{\infty}(X))$ is extremely disconnected.
Proof. By Corollary 22.6.8, $L^{\infty}(X; \real)$ is order complete. By Corollary 34.6.4, $\Omega(L^{\infty}(X))$ is extremely disconnected.$\square$
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