Definition:Closed Rectangle

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 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 the case where $a_i > b_i$ for some $i$, the closed rectangle $\ds \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$ 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.

Notation

A convenient notation for the closed rectangle $\ds \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$ is $\sqbrk {\closedint {\mathbf a} {\mathbf b} }$.

Also see

• Definition:Open Rectangle
• Definition:Half-Open Rectangle

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): closed interval
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): interval
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): closed interval
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): interval