= Discrete Stone-Čech compactifications are extremally disconnected
For a <discrete space> $X$ and $D\subseteq X$, the continuous extension of $\mathbf1_D$ 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.
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