The word dimensionMathworldPlanetmathPlanetmath in mathematics has many definitions, but all of them are trying to quantify our intuition that, for example, a sheet of paper has somehow one less dimension than a stack of papers.

One common way to define dimension is through some notion of a number of independent quantities needed to describe an element of an object. For example, it is natural to say that the sheet of paper is two-dimensional because one needs two real numbers to specify a position on the sheet, whereas the stack of papers is three-dimension because a position in a stack is specified by a sheet and a position on the sheet. Following this notion, in linear algebra the http://planetmath.org/Dimension2dimension of a vector space is defined as the minimal number of vectors such that every other vector in the vector space is representable as a sum of these. Similarly, the word rank denotes various dimension-like invariants that appear throughout the algebraMathworldPlanetmath.

However, if we try to generalize this notion to the mathematical objects that do not possess an algebraic structurePlanetmathPlanetmath, then we run into a difficulty. From the point of view of set theoryMathworldPlanetmath there are http://planetmath.org/Cardinalityas many real numbers as pairs of real numbers since there is a bijection from real numbers to pairs of real numbers. To distinguish a plane from a cube one needs to impose restrictionsPlanetmathPlanetmathPlanetmath on the kind of mapping. Surprisingly, it turns out that the continuity is not enough as was pointed out by Peano. There are continuous functionsMathworldPlanetmathPlanetmath that map a square onto a cube. So, in topologyMathworldPlanetmath one uses another intuitive notion that in a high-dimensional space there are more directions than in a low-dimensional. Hence, the (Lebesgue coveringPlanetmathPlanetmath) dimension of a topological space is defined as the smallest number d such that every covering of the space by open sets can be refined so that no point is contained in more than d+1 sets. For example, no matter how one covers a sheet of paper by sufficiently small other sheets of paper such that two sheets can overlap each other, but cannot merely touch, one will always find a point that is covered by 2+1=3 sheets.

Another definition of dimension rests on the idea that higher-dimensional objects are in some sense larger than the lower-dimensional ones. For example, to cover a cube with a side length 2 one needs at least 23=8 cubes with a side length 1, but a square with a side length 2 can be covered by only 22=4 unit squares. Let N(ϵ) be the minimal number of open ballsPlanetmathPlanetmath in any covering of a bounded set S by balls of radius ϵ. The http://planetmath.org/HausdorffDimensionBesicovitch-Hausdorff dimension of S is defined as -limϵ0logϵN(ϵ). The Besicovitch-Hausdorff dimension is not always defined, and when defined it might be non-integral.

Title dimension
Canonical name Dimension
Date of creation 2013-03-22 14:02:50
Last modified on 2013-03-22 14:02:50
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 10
Author bbukh (348)
Entry type Topic
Classification msc 00-01
Classification msc 15A03
Classification msc 54F45
Related topic Dimension
Related topic Dimension2
Related topic DimensionKrull
Related topic HausdorffDimension