The set $ C= C ( X) $ of all continuous, real valued functions on a topological space$ X$ will be provided with an algebraic structure and an order structure.
zero -set means:
$ Z(f ) = f^{-1}(\{ 0 \} ) = \{ x \in X \mid f ( x ) = 0 \} \quad ( f \in C ( X ) ) $
Thus, the open sets
$ pos( f ) = \{ x \mid f(x ) > 0 \}$
$ neg( f ) = \{ x \mid f(x ) < 0 \}$
are cozero-sets.
a space $X$ is said to be extremely disconnected if every open set has an open closure. $X$ is basically disconnected if every cozero-set has an open closure. hence any extremely disconnected space is basically disconnected. The converse fails.
My questions :
1: $X$ is extremely disconnected if and only if every pair of disjoint open sets have disjoint closures. What is the analogous condition for basically disconnected spaces?
2:In an basically disconnected space, any two disjoint open sets are completely seperated. In a basically disconnected spaces , any two dijoint cozero-sets are completely separated; equivalently , for every $ f \in C(X)$ , pos $f$ and neg $f$ are completely separated?