 Discrete Stone-Čech compactifications are extremally disconnected

For a discrete space $X$ and $D\subseteq X$, the continuous extension of $1_D$ to $\beta X$ remains idempotent by density, so it takes values in $\{0,1\}$. Its one-set is the clopen closure of $D$. If $O$ is any open set in $\beta X$, density gives $\overline{O}=\overline{O\cap X}$, so this closure is clopen as well. Thus $\beta X$ is an extremally disconnected space. For infinite $X$, this does not imply that every subset of $\beta X$ has open closure.