Cartesian products

There is a convenient way to construct new topological spaces from existing ones. Suppose that $ X$ and $ Y$ are topological spaces. The Cartesian product, $ X \times Y$, defines a new topological space as follows. Every $ x \in X$ and $ y \in Y$ generates a point $ (x,y)$ in $ X \times Y$. Each open set in $ X \times Y$ is formed by taking the Cartesian product of one open set from $ X$ and one from $ Y$. Exactly one open set exists in $ X \times Y$ for every pair of open sets that can be formed by taking one from $ X$ and one from $ Y$. Furthermore, these new open sets are used as a basis for forming the remaining open sets of $ X \times Y$ by allowing any unions and finite intersections of them.

A familiar example of a Cartesian product is $ {\mathbb{R}}\times {\mathbb{R}}$, which is equivalent to $ {\mathbb{R}}^2$. In general, $ {\mathbb{R}}^n$ is equivalent to $ {\mathbb{R}}
\times {\mathbb{R}}^{n-1}$. The Cartesian product can be taken over many spaces at once. For example, $ {\mathbb{R}}\times {\mathbb{R}}\times \cdots \times {\mathbb{R}}
= {\mathbb{R}}^n$. In the coming text, many important manifolds will be constructed via Cartesian products.

Steven M LaValle 2012-04-20