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 $\mathbb^2$ such that the sides of the rectangles are parallel to the coordinate axes. Every $k$-cell is compact.

Formal definition

For every integer $i$ from $1$ to $k$, let $a_i$ and $b_i$ be real numbers such that for all $a_i < b_i$. The set of all points $x=(x_1,\dots,x_k)$ in $\mathbb^k$ whose coordinates satisfy the inequalities $a_i\leq x_i\leq b_i$ is a $k$-cell.

Intuition

A $k$-cell of dimension $k\leq 3$ is especially simple. For example, a 1-cell is simply the interval ,b/math> 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.

Notes

References

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* {{cite book, first=Walter, last=Rudin, author-link1=Walter Rudin, title= Principles of Mathematical Analysis, year=1976, publisher=McGraw-Hill

Basic concepts in set theory
Compactness (mathematics)