Definition:Cartesian Product/Cartesian Space

Definition
Let $S$ be a set.

Then the cartesian $n$th power of $S$, or $S$ to the power of $n$, is defined as:


 * $\displaystyle S^n = \prod_{k \mathop = 1}^n S = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S}\right\}$

Thus $S^n = S \times S \times \ldots \left({n}\right) \ldots \times S$

Alternatively it can be defined recursively:


 * $S^n = \begin{cases}

S: & n = 1 \\ S \times S^{n-1} & n > 1 \end{cases}$

The set $S^n$ called a cartesian space.

An element $x_j$ of a tuple $\left({x_1, x_2, \ldots, x_n}\right)$ of a cartesian space $S^n$ is known as a basis element of $S^n$.

Real Cartesian Space
When $S$ is the set of real numbers $\R$, the cartesian product takes on a special significance.

Let $n \in \N^*$.

Then $\R^n$ is the cartesian product defined as follows:


 * $\displaystyle \R^n = \R \times \R \times \cdots \left({n}\right) \cdots \times \R = \prod_{k \mathop = 1}^n \R$

Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:


 * $\R^n = \left\{{\left({x_1, x_2, \ldots, x_n}\right): x_1, x_2, \ldots, x_n \in \R}\right\}$

Also see
It can be shown that:
 * $\R^2$ is isomorphic to any infinite flat plane in space
 * $\R^3$ is isomorphic to the whole of space itself.

See the definition of a Real Vector Space.