Generic filter

Formally, let P be a partially ordered set, and let F be a filter on P; that is, F is a subset of P such that:

1. F is nonempty
2. If p, q ∈ P and p ≤ q and p is an element of F, then q is an element of F (F is closed upward)
3. If p and q are elements of F, then there is an element r of F such that r ≤ p and r ≤ q (F is downward directed)

Now if D is a collection of dense open subsets of P, in the topology whose basic open sets are all sets of the form {q | q ≤ p} for particular p in P, then F is said to be D-generic if F meets all sets in D; that is,

${\displaystyle F\cap E\ne \varnothing ,}$ for all E ∈ D.

Similarly, if M is a transitive model of ZFC (or some sufficient fragment thereof), with P an element of M, then F is said to be M-generic, or sometimes generic over M, if F meets all dense open subsets of P that are elements of M.