When the continuum has cofinality ω₁

In this paper we consider models of set theory in which the continuum has cofinality ω₁ We show that it is consistent with CH that for any complete boolean algebra B of cardinality less than or equal to c (continuum) there exists an ω₁-generated ideal J in P(ω) (power set of ω) such that B is isomorphic to P(ω)mod J. We also show that the existence of generalized Luzin sets for every ω₁-saturated ideal in the Borel sets does not imply Martin’s axiom.