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:

$\displaystyle \prod_{i \mathop = 1}^n \openint {a_i} {b_i} = \openint {a_1} {b_1} \times \cdots \times \openint {a_n} {b_n} \subseteq \R^n$

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 $\displaystyle \prod_{i \mathop = 1}^n \openint {a_i} {b_i}$ as a convenient abbreviation.


Also see


Sources