# Definition:Half-Open Rectangle

## Definition

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

The set:

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

is called an **half-open rectangle in $\R^n$** or **half-open $n$-rectangle**.

Here, $\times$ denotes Cartesian product.

The collection of all **half-open $n$-rectangles** is denoted $\JJ_{ho}$, or $\JJ_{ho}^n$ if the dimension $n$ is to be emphasized.

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 $\horectr {\mathbf a} {\mathbf b}$ for $\ds \prod_{i \mathop = 1}^n \hointr {a_i} {b_i}$ as a convenient abbreviation.

Of course, sets of the form $\horectl {\mathbf a} {\mathbf b}$ have equal right to be called **half-open rectangles**, but these are rarely encountered.

## Also see

- Definition:Open Rectangle
- Definition:Half-Open Real Interval, the special case such that $n = 1$

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*: $\S 3$