What is "product topology"

n. (context topology English) The topology of the Cartesian product of two or more topological spaces which is generated from a basis whose elements are Cartesian products of open set subsets of the original spaces, such that for each such Cartesian product only a finite quantity of such subsets are proper subset.

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural".