“K-cell (mathematics)”的意思、由来-开放百科全书

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K-cell (mathematics)

释义

Formal definition
Intuition
References

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.^[1] 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 {{nowrap|1=R²}} such that the sides of the rectangles are parallel to the coordinate axes.

Formal definition

Let a_i ∈ {{nowrap|1=R}} and b_i ∈ {{nowrap|1=R}}. If a_i < b_i for all i = 1,...,k, the set of all points x = (x₁,...,x_k) in {{nowrap|1=R^k}} whose coordinates satisfy the inequalities a_i ≤ x_i ≤ b_i is a k-cell.^[2]

Every k-cell is compact.^[3]

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.

References

1. ^{{cite book|last=Foran|first=James|title=Fundamentals of Real Analysis|url=https://books.google.com/books?id=sDjz8x0hJ44C&pg=PA24|accessdate=23 May 2014|date=1991-01-07|publisher=CRC Press|isbn=9780824784539|pages=24–}}
2. ^Rudin, W: Principles of Mathematical Analysis, page 31. McGraw-Hill, 1976.
3. ^Rudin, W: Principles of Mathematical Analysis, page 39. McGraw-Hill, 1976.

2 : Basic concepts in set theory|Compactness (mathematics)

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