Bell writes on page 21 (you may use the search in the preview to search for "21" to view the page):
"..., we show that, for any complete Boolean algebra $B$, all the theorems of $ZFC$ are true in $V^{(B)}$. ..."
On Wikipedia it says: "For any poset $P$ there is a complete Boolean algebra $B$ and a map $e$ from $P$ to $B^+$ (the non-zero elements of $B$) such that the image is dense, $e(p)\le e(q)$ whenever $p \le q$, and $e(p)e(q)=0$ whenever $p$ and $q$ are incompatible. This Boolean algebra is unique up to isomorphism. It can be constructed as the algebra of regular open sets in the topological space of $P$ (with underlying set $P$, and a base given by the sets $U_p$ of elements $q$ with $q\le p$)."
Question 1: Is it the case that if we extend $V$ then we can use any Boolean algebra but if we use a model $M \subset V$ then we have to use the unique Boolean algebra as described on Wikipedia?
Question 2: Assume I have a model $M$ of some theory, not necessarily $ZF$, and I want to extend it. Then I first want to construct a Boolean valued model. To this end, I first want to construct a suitable Boolean algebra satisfying the requirements mentioned on Wikipedia. (Is this correct so far?) Then what do regular open sets look like in the topology generated by $U_p$? (A set is called regular open if $U = \mathring{\overline{U}}$)
Many thanks for your help.