dimension

The word dimension in mathematics has many definitions, butall of them are trying to quantify our intuition that, forexample, a sheet of paper has somehow one less dimension than astack of papers.

One common way to define dimension is through some notion of anumber of independent quantities needed to describe an elementof an object. For example, it is natural to say that the sheet ofpaper is two-dimensional because one needs two real numbers tospecify a position on the sheet, whereas the stack of papers isthree-dimension because a position in a stack is specified by a sheetand a position on the sheet. Following this notion, in linearalgebra the http://planetmath.org/Dimension2dimension of a vector spaceis defined as the minimal number of vectors such that every othervector in the vector space is representable as a sum of these.Similarly, the word rank denotes various dimension-likeinvariants that appear throughout the algebra.

However, if we try to generalize this notion to the mathematicalobjects that do not possess an algebraic structure, then we runinto a difficulty. From the point of view of set theory there arehttp://planetmath.org/Cardinalityas many real numbers as pairs of realnumbers since there is a bijection from real numbers to pairs ofreal numbers. To distinguish a plane from a cube one needs toimpose restrictions on the kind of mapping. Surprisingly, it turnsout that the continuity is not enough as was pointed out by Peano.There are continuous functions that map a square onto a cube. So,in topology one uses another intuitive notion that in ahigh-dimensional space there are more directions than in alow-dimensional. Hence, the (Lebesgue covering) dimension of atopological space is defined as the smallest number $d$ such thatevery covering of the space by open sets can be refined so that nopoint is contained in more than $d+1$ sets. For example, no matterhow one covers a sheet of paper by sufficiently small other sheetsof paper such that two sheets can overlap each other, butcannot merely touch, one will always find a point that is coveredby $2+1=3$ sheets.

Another definition of dimension rests on the idea thathigher-dimensional objects are in some sense larger than thelower-dimensional ones. For example, to cover a cube with a sidelength $2$ one needs at least $2^3=8$ cubes with a side length$1$, but a square with a side length $2$ can be covered by only$2^2=4$ unit squares. Let $N(ϵ)$ be the minimal number ofopen balls in any covering of a bounded set $S$ by balls of radius$ϵ$. The http://planetmath.org/HausdorffDimensionBesicovitch-Hausdorff dimension of $S$ is definedas $-{lim}_{ϵ\to 0}{\log }_{ϵ}N(ϵ)$. TheBesicovitch-Hausdorff dimension is not always defined, and whendefined 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