  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''.
