Definition:Half-Open Rectangle

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Definition

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

The set:

$\displaystyle \prod_{i \mathop = 1}^n \left[{a_i \,.\,.\, b_i}\right) = \left[{a_1 \,.\,.\, b_1}\right) \times \cdots \times \left[{a_n \,.\,.\, b_n}\right) \subseteq \R^n$

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 $\mathcal{J}_{ho}$, or $\mathcal{J}_{ho}^n$ 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 $\varnothing$.

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 $\left[\left[{\mathbf a \,.\,.\, \mathbf b}\right)\right)$ for $\displaystyle \prod_{i \mathop = 1}^n \left[{a_i \,.\,.\, b_i}\right)$ as a convenient abbreviation.

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


Also see


Sources