  Using the Feferman-Vaught Theorem, we prove that a definable subset of a
product structure must be a Boolean combination of open sets, in the product
topology induced by giving each factor structure the discrete topology. We
prove a converse of the Feferman-Vaught theorem for families of structures with
certain properties, including families of integral domains. We use these
results to obtain characterizations of the definable subsets of $\prod_p
\mathbb{F}_p$ -- in particular, every formula is equivalent to a Boolean
combination of $\exists \forall \exists$ formulae.
