Rasiowa–Sikorski lemma

The Rasiowa–Sikorski lemma says that in a countable partially ordered set, a filter can meet every set in any countable family of dense subsets.

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Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. The lemma’s conditions and conclusion are expressed using an order on a countable set.
Partial order: A reflexive, antisymmetric, transitive relation on a set. The lemma applies to sets whose elements are ordered by a partial order.
Forcing (mathematics): A method in set theory that builds models by adjoining generic objects through partially ordered conditions. The lemma is a basic tool for producing filters that meet dense requirements in forcing orders.
Helena Rasiowa: A Polish mathematician whose work helped develop algebraic logic and Boolean-valued methods. Her name is attached to the lemma arising from work on Boolean-valued constructions.
Rasiowa–Sikorski lemma for uncountable families: The question of whether a filter can meet an uncountable family of dense sets in a partially ordered set. The lemma guarantees this only for countable families; larger families require additional hypotheses.
Dense subset: A subset of a partially ordered set such that every element has a stronger element in the subset. Each target set can be reached from any current condition in the construction.
Forcing condition: An element of a forcing partial order representing a piece of information in a possible generic extension. Forcing commonly uses the lemma’s ordered sets as collections of conditions.
Generic filter: A filter on a forcing partial order that meets every dense set in a specified model. Meeting a chosen countable family of dense sets is the lemma’s central construction.
Roman Sikorski: A Polish mathematician known for contributions to Boolean algebras, measure theory, and mathematical logic. His work with Rasiowa is associated with the result’s formulation and use.
Axiom of Dependent Choice: An axiom asserting the existence of infinite sequences of successive choices when each choice depends on its predecessor. The lemma’s sequence construction illustrates dependent choice, while its countability and order assumptions give a specific conclusion.