k-cell (mathematics)

A ${\displaystyle k}$-cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of ${\displaystyle k}$ closed intervals on the real line. This means that a ${\displaystyle k}$-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. The ${\displaystyle k}$ intervals need not be identical. For example, a 2-cell is a rectangle in ${\displaystyle \mathbb {R} ^{2}}$ such that the sides of the rectangles are parallel to the coordinate axes. Every ${\displaystyle k}$-cell is compact.