Definition:Open Rectangle

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

The set:


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

is called an open rectangle in $\R^n$ or open $n$-rectangle; here, $\times$ denotes Cartesian product.

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

Alternative Notation
Some authors write $\left(({a .. b}\right))$ for $\displaystyle \prod_{i=1}^n \left({a_i .. b_i}\right)$ as a convenient abbreviation.