  In the literature, there is no known general method (formula) to compute the
Zariski closure of an ``infinite'' subset of the prime spectrum. This problem
indeed deals with the prime ideals of an infinite direct product of nonzero
commutative rings that are very complicated to understand (the structure of
most of them is unknown). In this article, by appealing to the patch closure
and using laying over minimal prime technique, we overcome the above obstacle
and then obtain new and quite useful results for computing the Zariski and flat
closures of an infinite subset of the prime spectrum.
