  Every ordered collection of sets in Euclidean space can be associated to a
combinatorial code, which records the regions cut out by the sets in space.
Given two ordered collections of sets, one can form a third collection in which
the $i$-th set is the Cartesian product of the corresponding sets from the
original collections. We prove a general "product theorem" which characterizes
the code associated to the collection resulting from this operation, in terms
of the codes associated to the original collections. We use this theorem to
characterize the codes realizable by axis-parallel boxes, and exhibit
differences between this class of codes and those realizable by convex open or
closed sets. We also use our theorem to prove that a "monotonicity of open
convexity" result of Cruz, Giusti, Itskov, and Kronholm holds for closed sets
when some assumptions are slightly weakened.
