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 Now we can state the Rasiowa–Sikorski lemma: Let p ∈ P be given.
Since D is countable, D = { Di | i ∈ N }, i.e., one can enumerate the dense subsets of P as D1, D2, ... and, by density, there exists p1 ≤ p with p1 ∈ D1.
The Rasiowa–Sikorski lemma can be viewed as an equivalent to a weaker form of Martin's axiom.
More specifically, it is equivalent to MA(ℵ0) and to the axiom of countable choice.