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 2020-08-14