Editing Interval (mathematics) (section)

=== Multi-dimensional intervals ===

A finite interval is (the interior of) a 1-dimensional [[hyperrectangle]]. Generalized to [[real coordinate space]] <math>\R^n,</math> an [[axis-aligned object|axis-aligned]] hyperrectangle (or box) is the [[Cartesian product]] of <math>n</math> finite intervals. For <math>n=2</math> this is a [[rectangle]]; for <math>n=3</math> this is a [[rectangular cuboid]] (also called a "[[box (geometry)|box]]").

Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any <math>n</math> intervals, <math>I = I_1\times I_2 \times \cdots \times I_n</math> is sometimes called an '''<math>n</math>-dimensional interval'''.{{cn|date=September 2023}}

A '''facet''' of such an interval <math>I</math> is the result of replacing any non-degenerate interval factor <math>I_k</math> by a degenerate interval consisting of a finite endpoint of <math>I_k.</math> The '''faces''' of <math>I</math> comprise <math>I</math> itself and all faces of its facets. The '''corners''' of <math>I</math> are the faces that consist of a single point of <math>\R^n.</math>{{cn|date=September 2023}}