Definition:Closed Rectangle
extract degenerate case
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Definition
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:
- $\ds \prod_{i \mathop = 1}^n \closedint {a_i} {b_i} = \closedint {a_1} {b_1} \times \cdots \times \closedint {a_n} {b_n} \subseteq \R^n$
is called a closed rectangle in $\R^n$ or closed $n$-rectangle.
Degenerate Case
In case $a_i > b_i$ for some $i$, the closed 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
A convenient abbreviation is $\sqbrk {\closedint {\mathbf a} {\mathbf b} }$ for $\ds \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$.