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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 ''D'' of a forcing notion (''P'', ≤) is called dense in ''P'' if for any ''p'' ∈ ''P'' there is ''d'' ∈ ''D'' with ''d'' ≤ ''p''. 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 family of dense subsets of ''P'' then there exists a ''D''-generic filter ''F'' in ''P'' such that ''p'' ∈ ''F''. == Proof of the Rasiowa–Sikorski lemma == The proof runs as follows: since ''D'' is countable, one can enumerate the dense subsets of ''P'' as ''D''1, ''D''2, …. By assumption, there exists ''p'' ∈ ''P''. Then by density, there exists ''p''1 ≤ ''p'' with ''p''1 ∈ ''D''1. Repeating, one gets … ≤ ''p''2 ≤ ''p''1 ≤ ''p'' with ''p''''i'' ∈ ''D''''i''. Then ''G'' = is a ''D''-generic filter. The Rasiowa–Sikorski lemma can be viewed as a weaker form of an equivalent to Martin's axiom. More specifically, it is equivalent to MA(). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rasiowa–Sikorski lemma」の詳細全文を読む スポンサード リンク
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