Hyperrectangles

Why

We want to generalize rectangles and cubes to $n$-dimensional space.

Definition

Let $I: \upto{d} \to \R $ be a family of $d$ intervals. A hyperrectangle is the set $\prod_{i=1}^{d} I_i$.¹

As a result of this definition, an interval, a rectangle, and a cube are all hyperrectangles. Of course, in our definition we include four, five, and “dimensional” rectangles.

As with intervals, rectangles, and cubes, we call a hyperrectangle open, closed, left-open, right-open accordingly.

1. Some authors use the term rectangle or $n$-dimensional rectangle. Some authors use the term box or $n$-box. ↩︎