Rasiowa–Sikorski lemma

In axiomatic set theory, the Rasiowa–Sikorski lemma named after Helena Rasiowa and Roman Sikorski is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset E of a poset (P, ≤) is called dense in P if for any p ∈ P there is e ∈ E with e ≤ p. If D is a set of dense subsets of P, then a filter F in P is called D-generic if

F ∩ E ≠ ∅ for all E ∈ D.

Now we can state the Rasiowa–Sikorski lemma:

Let (P, ≤) be a poset and p ∈ P. If D is a countable set of dense subsets of P then there exists a D-generic filter F in P such that p ∈ F.
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