Definition:Closed 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 of closed intervals:

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

is called a closed rectangle in $\R^n$ or closed $n$-rectangle.

Degenerate Case

In case $a_i > b_i$ 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

A convenient abbreviation is $\left[[{\mathbf a \,.\,.\, \mathbf b}\right]]$ for $\displaystyle \prod_{i \mathop = 1}^n \left[{a_i \,.\,.\, b_i}\right]$.

Also see