Basically and extremely disconnected spaces

Question

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?

Answer 1

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?

A naive guess is that a space $X$ is basically disconnected if and only if every pair of its disjoint cozero-sets have disjoint closures. But I can prove only “only if” part. Let $X$ be a basically disconnected space and $A$ and $B$ be its disjoint cozero-sets. Since the set $B$ is open, sets $B$ and $\overline{A}$ are disjoint. Since the set $\overline{A}$ is open, sets $\overline{B}$ and $\overline{A}$ are disjoint.

2: In a basically disconnected spaces , any two disjoint cozero-sets are completely separated; equivalently , for every $ f \in C(X)$ , pos $f$ and neg $f$ are completely separated?

Yes. The sets $pos(f)$ and $neg(f)$ are disjoint co-zero sets. Hence the sets $\overline{pos(f)}$ and $\overline{neg(f)}$ are disjoint open and closed sets, so they are completely separated by a function $g:X\to [0,1]$ such that $g(x)=0$ for each $x\in \overline{neg(f)}$, $g(x)=1$ for each $x\in \overline{pos(f)}$, and $g(x)=1/2$, otherwise. I recall that two subsets $A$ and $B$ of a topological space are called completely separated if there exists a continuous function $g:X\to [0,1]$ such that $g(x)=0$ for each $x\in A$ and $g(x)=1$ for each $x\in B$.