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

  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 $x \mapsto f(x) / (f(x) + g(x))$.

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