$X$ is basically disconnected if every cozero-set has an open closure.

a space $X$ is said to be extremally disconnected if every open set has an open closure.

$X$ is basically disconnected if every cozero-set has an open closure.

hence any extremally disconnected space is basically disconnected. The converse fails.

1: Is every open subspace of an extermally disconnected space extermally disconnected? Is it true for basically disconnected?

2:In an extermally disconnected space, are any two disjoint zero-sets completely seperated?

( Or even in an bacically disconnected space, are any two disjoint cozero-sets completely seperated?)

1 Answer

1. Yes for extremally disconnected, and the proof is really straightforward, so you should think about it. Not necessarily for basically disconnected I think. A cozero set in the subspace does not have to be the intersection of a cozero set in the whole space and the subspace. It is if the open subspace is $$C^*$$-embedded or if the subspace itself is cozero in the whole space. I would take a basically connected space that is not extremally connected and try to find a counterexample.
2. Every two disjoint zero sets are completely separated and this has nothing to do with extremal or basic disconnectedness. If $$A = f^{-1}(0)$$ and $$B = g^{-1}(0)$$ and they are disjoint, consider $$x \mapsto f(x) / (f(x) + g(x))$$.