Is it true that a dense subset of a complete Boolean algebra has supremum 1?

Question

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$ ?

Answer 1

Suppose $\bigvee_{a \in S} a = b < 1$. Then in particular every element of $S$ is $\leq b$, so no nonzero element of $S$ is $\leq b^c$, because $b \wedge b^c = 0$. So $S$ is not dense.

Answer 2

Yes, the join of any dense subset of a Boolean algebra is $1$, because otherwise it would be bounded by some $b$ less than $1$, and so $\neg b$ would have no elements of the dense set below it, contradicting density.

There is no need to assume that the Boolean algebra is complete.