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.

istrue is that a clopen subset of an extremally disconnected set is extremally disconnected (straight from the definition). Likewise the quotient of a complete Boolean algebra by a principal ideal is complete, and the homomorphism is a complete homomorphism. $\endgroup$