There is a convenient way to construct new topological spaces from existing ones. Suppose that and are topological spaces. The Cartesian product, , defines a new topological space as follows. Every and generates a point in . Each open set in is formed by taking the Cartesian product of one open set from and one from . Exactly one open set exists in for every pair of open sets that can be formed by taking one from and one from . Furthermore, these new open sets are used as a basis for forming the remaining open sets of by allowing any unions and finite intersections of them.
A familiar example of a Cartesian product is , which is equivalent to . In general, is equivalent to . The Cartesian product can be taken over many spaces at once. For example, . In the coming text, many important manifolds will be constructed via Cartesian products.
Steven M LaValle 2012-04-20