Let $S\subseteq B$ be a dense subset of a complete Boolean algebra $B$. Is is true that $\sum S = 1$? Jech seems to use this fact several times in his book (e.g. the proof of 7.15) but I have been unable to prove it, if it is true.
Is it true if we tighten the condition to open dense, i.e. whenever non-zero $u \le v$ for $v \in S$, then $ u \in S$ ?