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 a 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.

Degenerate Case

In the case where $a_i > b_i$ for some $i$, the half-open rectangle $\ds \prod_{i \mathop = 1}^n \hointr {a_i} {b_i}$ degenerates to 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 half-open rectangle $\ds \prod_{i \mathop = 1}^n \hointr {a_i} {b_i}$ is $\horectr {\mathbf a} {\mathbf b}$.

Also defined as

Sets of the form:

$\horectl {\mathbf a} {\mathbf b} \:= \ds \prod_{i \mathop = 1}^n \hointl {a_i} {b_i}$

can equally well be called half-open rectangles as those of the form $\horectr {\mathbf a} {\mathbf b}$.

However, these are rarely encountered.

Also see

• Definition:Open Rectangle
• Definition:Closed Rectangle

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

• Results about half-open rectangles can be found here.

Technical Note

The $\LaTeX$ code for \(\horectr {\mathbf a} {\mathbf b}\) is \horectr {\mathbf a} {\mathbf b} .

This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation and its derivatives.

The name is derived from half-open rectangle on the right.

Sources

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