K-cell (mathematics)

A "k"-cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of "k" closed intervals on the real line. This means that a "k"-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. The "k" intervals need not be identical. For example, a 2-cell is a rectangle in such that the sides of the rectangles are parallel to the coordinate axes.
Formal definition.
Let "a""i" ∈ and "b""i" ∈ . If "a""i" < "b""i" for all "i" = 1...,"k", the set of all points x = ("x"1...,"x""k") in whose coordinates satisfy the inequalities "a"i ≤ "x""i" ≤ "b""i" is a "k"-cell.
Every "k"-cell is compact.
Intuition.
A "k"-cell of dimension "k" ≤ 3 is especially simple. For example, a 1-cell is simply the interval ["a","b"] with "a" < "b". A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid. 
The sides and edges of a "k"-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells.