For any family F of sets, let
ℬ(F) denote the smallest σalgebra containing F. Throughout this paper
X denotes a set and ℛ the family of sets of the form A × B, for A ⊆ X
and B ⊆ X. It is of interest to find conditions under which the following
holds:
(1) Each subset of X × X is a member of ℬ(ℛ_{−})
The interesting case is when ω_{1} < Card X ≦ c, since results for other cases are
known. It is shown in Theorem 9 that (1) is equivalent to
(2) There is a countable ordinal α such that each subset of X × X oan be
generated from ℛ is α Baire process steps.
It is also shown that the twodimensional statements (1) and (2) are equivalent to
the onedimensional statement
(3) There is a countable ordinal α such that for each family H of subsets of X
with Card H = Card X, there is a oountable family G such that each member of H
can be generated from G in α steps.
It is shown in Theorem 5 that the continuum hypothesis (CH) is
equivalent to certain statements about rectangles of the form (1) and (2) with
α = 2.
