Embeddings of Cohen algebras

Abstract

Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space {0,1}^kappa equipped with product measure, and C(kappa), the algebra of regular open sets in the space {0,1}^kappa, for kappa an infinite cardinal. C(kappa) is much easier to analyse than B(kappa) : C(kappa) has a dense subset of size kappa, while the density of B(kappa) depends on the cardinal characteristics of the real line; and the definition of C(kappa) is simpler. Indeed, C(kappa) seems to have the simplest definition among all algebras of its size. In the Main Theorem of this paper we show that in a certain precise sense, C(aleph_1) has the simplest structure among all algebras of its size, too. MAIN THEOREM: If ZFC is consistent then so is ZFC + 2^{aleph_0}= aleph_2 +``for every complete Boolean algebra B of uniform density aleph_1, C(aleph_1) is isomorphic to a complete subalgebra of B''.