Solution
= Solution
A set $G\subseteq\mathbb P$ is a filter when it is upward closed toward weaker conditions and downward directed: if $p\in G$ and $p\leq q$, then $q\in G$, while any $p,q\in G$ have some $r\in G$ with $r\leq p,q$. It is a <generic filter> over $M$ when
$$
\boxed{G\cap D\ne\varnothing\quad\text{for every dense }D\subseteq\mathbb P\text{ with }D\in M.}
$$