### Embeddings of Cohen algebras

by Shelah and Zapletal. [ShZa:561]

Advances in Math, 1997

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''.

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