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.