TITLE: Stone-Čech compactification
QUESTION [5 upvotes]: Is every hyperstonean space a Stone-Čech compactification of a discrete space?
Is there a closed subset of Stone-Čech boundary that is extremally disconnected?

REPLY [5 votes]: As for Q1, the answer is no. As remarked by Narutaka, the hyperstonean cover of $[0,1]$ does not have isolated points (Corollary 2.22) and all points of a discrete space $\Gamma$ are isolated in $\beta \Gamma$. Hyperstonean spaces can be way more wilder; all relevant information can be found in the book Banach spaces of continuous functions as dual spaces by Dales, Dashiell, Lau, and Strauss.
As for Q2, if you mean $\beta \mathbb N \setminus \mathbb N$, then yes, there are lots of copies of $\beta \mathbb N$ therein.