Power set modulo small, the singular of uncountable cofinality

Abstract Let μ be singular of uncountable cofinality. If μ > 2cf(μ), we prove that in ℙ = ([μ]μ, ⊇) as a forcing notion we have a natural complete embedding of Levy(ℵ0, μ+) (so ℙ collapses μ+to ℵ0) and even Levy ( ). The “natural” means that the forcing ({p∈ [μ] :pclosed}, ⊇) is naturally embedded and is equivalent to the Levy algebra. Also if ℙ fails theχ-c.c. then it collapsesχto ℵ0(and the parallel results for the case μ > ℵ0is regular or of countable cofinality). Moreover we prove: for regular uncountableκ, there is a familyPof κpartitionsĀ= ⟨Aα:α<κ⟩ ofκsuch that for anyA∈ [κ]κfor some ⟨Aα:α<κ⟩ ∈Pwe have α <κ⇒ ∣Aα∩A∣ =κ.