Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset.
Question: Is $S$ extremally disconnected?
For me, this looks like a very natural question about extremally disconnected sets. However, on the spot, I could neither prove this, nor find a counterexample. Also, I was not able to find anything on this in the literature.
Best Answer
No, the Stone-Cech compactification $\beta\mathbf{N}$ of $\mathbf{N}$ is extremally disconnected, but not the Stone-Cech boundary $\beta\mathbf{N}\smallsetminus\mathbf{N}$.
To see this, it is enough to find an increasing sequence $(F_n)$ of clopen subsets with no supremum (=least upper bound) in the Boolean algebra of clopen subsets, or equivalently of the Boolean algebra of subsets modulo symmetric difference finite subsets.
For this, it is more convenient to work with $\mathbf{N}^2$: then $F_n$ is just the boundary of $\mathbf{N}\times\{0,\dots,n\}$. An upper bound for this sequence is just a subset $Y$ intersecting each horizontal line in a cofinite subset. By given any such $Y$ one can remove one point in each horizontal layer and get another upper bound $Y'\subset Y$ with $Y\smallsetminus Y'$ infinite. So there is no least upper bound.