Definition:Open Rectangle

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Let $n\geq1$ be a natural number.

Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers.

The Cartesian product:

$\displaystyle \prod_{i \mathop = 1}^n \left({a_i \,.\,.\, b_i}\right) = \left({a_1 \,.\,.\, b_1}\right) \times \cdots \times \left({a_n \,.\,.\, b_n}\right) \subseteq \R^n$

is called an open rectangle in $\R^n$ or open $n$-rectangle.

The collection of all open $n$-rectangles is denoted $\mathcal{J}_o$, or $\mathcal{J}_o^n$ 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 $\varnothing$.

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 $\left(({\mathbf a \,.\,.\, \mathbf b}\right))$ for $\displaystyle \prod_{i \mathop = 1}^n \left({a_i \,.\,.\, b_i}\right)$ as a convenient abbreviation.

Also see