\begin{definition}[Definition:Open Rectangle]
Let $n \ge 1$ be a natural number.
Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers.
The Cartesian product:
:$\ds \prod_{i \mathop = 1}^n \openint {a_i} {b_i} = \openint {a_1} {b_1} \times \cdots \times \openint {a_n} {b_n} \subseteq \R^n$
is called an '''open rectangle in $\R^n$''' or '''open $n$-rectangle'''.
The collection of all '''open $n$-rectangles''' is denoted $\JJ_o$, or $\JJ_o^n$ if the dimension $n$ is to be emphasized.
\end{definition}