# Definition:Open Rectangle

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## Definition

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.

### Degenerate Case

In case $a_i \ge b_i$ for some $i$, the rectangle is taken to be the empty set $\O$.

This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products.

## Also known as

Some authors write $\paren {\openint {\mathbf a} {\mathbf b} }$ for $\ds \prod_{i \mathop = 1}^n \openint {a_i} {b_i}$ as a convenient abbreviation.

## Also see

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**open interval** - 2005: René L. Schilling:
*Measures, Integrals and Martingales*: $\S 3$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**open interval**