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k-cell (mathematics)

From Wikipedia, the free encyclopedia
Projections of K-cells onto the plane (from to ). Only the edges of the higher-dimensional cells are shown.

A -cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of closed intervals on the real line.[1] This means that a -dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. The 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. Every -cell is compact.[2][3]

Formal definition

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For every integer from to , let and be real numbers such that for all . The set of all points in whose coordinates satisfy the inequalities is a -cell.[4]

Intuition

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A -cell of dimension is especially simple. For example, a 1-cell is simply the interval with . 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 -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

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References

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  • Foran, James (1991-01-07). Fundamentals of Real Analysis. CRC Press. ISBN 9780824784539. Retrieved 23 May 2014.
  • Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill.