\begin{definition}[Definition:Half-Open Rectangle]
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.
\end{definition}