I’m an undergraduate student trying to teach myself set-theory. And I have some trouble understanding the density of a constructed set.
In Lemma 10.1 of Nik’s book, it states: Let $G$ be a generic ideal of $P$, let $D \in \mathbf M$ be a subset of $P$, and suppose every element of $G$ is compatible with some element in $D$, then $G$ must intersect $D$.
In Nik’s proof, he introduced the set $$D’=\{p\in P:p\supset q\ \ {\rm for\ some\ }q\ {\rm in\ }D\}\cup\{p\in P:p\ {\rm is\ incompatible\ with\ every\ }q\in D\}$$ and claims that the set $D’$ is dense in $P$, which I had trouble verifying. The information I had seems so limited and I don’t know where to start with. I had to show that for any $q\in P$, there’s some $p\in D’$ such that $p$ extends $q$. I tried to assume $q\in D$, then it seems natural to divide it in to two cases, where
- There’s some $p\in D’$ such that $p$ extends $q$.
- Any $p\in D’$, $p$ is not a extension of $q$.
And I have to show that case 2 will not hold. But I can’t see how this could be done neither.
Can anyone help me with this problem? Am I missing or misunderstanding some basic ideas about this subject?
Given any condition in $p\in P$, either it is incompatible with the whole of $D$, in which case it is inside $D'$, or it is compatible with some $q\in D$, and therefore there is some $r$ which extends both $p$ and $q$, and therefore, by definition $r\in D'$.
So either $p$ was already in $D'$, or it has an extension in $D'$. In either case, $D'$ is dense. As a good exercise, try to prove that it is also open.