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.