Functional separation of regular open sets of a topological space

Question

Let $\langle X,\mathscr{O}\rangle$ be a topological space. We say that disjoint $A,B\in 2^X$ are functionally separated iff there exists a continuous function $f\colon X\to [0,1]$ such that:

• if $x\in A$, then $f(x)=0$
• if $x\in B$, then $f(x)=1$

We define $\langle X,\mathscr{O}\rangle$ to be completely regular iff for a closed set $C\in 2^X$ and a point $x\notin X$, $F$ and $\{x\}$ are functionally separated. Let $\langle\mathrm{r}\mathscr{O},+,\cdot,-,\emptyset,X\rangle$ be the complete boolean algebra of regular open subsets of completely regular space $\langle X,\mathscr{O}\rangle$. In this algebra define: \[ A\ll B\iff \mbox{$A$ and $-B$ are functionally separated.} \] What I am trying to prove are the following two properties of $\ll$:

1. if $A\ll C$ and $B\ll D$, then $A\cdot B\ll C\cdot D$
2. if $A\ll C$, then $\exists_{B\in 2^X\setminus\{\emptyset\}}\,A\ll B\ll C$

Concerning 1. suppose $f\colon X\to[0,1]$ and $g\colon X\to[0,1]$ are such that:

• $x\in A\rightarrow f(x)=0$ and $x\in B\rightarrow g(x)=0$
• $x\in -C\rightarrow f(x)=1$ and $x\in -D\rightarrow g(x)=1$

Now put $h(x):=\max\{f(x),g(x)\}$. Thus we have:

• $x\in A\cdot B\rightarrow x\in A\wedge x\in B\rightarrow f(x)=0 \wedge g(x)=0\rightarrow h(x)=0$
• $x\in -C+-D\rightarrow x\in\mathrm{Int}\,\mathrm{Cl}\, (-C\cup-D)\rightarrow\: ???$.

And I got stuck at the second dot above. Could you please give my any hint how to proceed?

Concerning 2. I would appreciate any suggestion.

Answer 1

For 1 note that $x ∈ \operatorname{Cl}(-C) ∪ \operatorname{Cl}(-D)$.

For 2 consider $B := f^{-1}[[0, 1/2)]$.

Answer 2

If $x\in -(C\cdot D)$, $$\begin{split}\Rightarrow & x \in \text{Cl}(-C)\text{ or }x\in \text{Cl}(-D) \\ \Rightarrow & f(x) =1\text{ or }g(x)=1 \\ \Rightarrow &h(x) = 1 \end{split}$$ as $f, g$ has range $[0,1]$).