# dimension

The word *dimension ^{}* 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 algebra^{}.

However, if we try to generalize this notion to the mathematical
objects that do not possess an algebraic structure^{}, then we run
into a difficulty. From the point of view of set theory^{} 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 restrictions^{} on the kind of mapping. Surprisingly, it turns
out 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 a
high-dimensional space there are more directions than in a
low-dimensional. Hence, the (Lebesgue covering^{}) 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 ${2}^{3}=8$ 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 $N(\u03f5)$ be the minimal number of
open balls^{} in any covering of a bounded set $S$ by balls of radius
$\u03f5$. The http://planetmath.org/HausdorffDimensionBesicovitch-Hausdorff dimension of $S$ is defined
as $-{lim}_{\u03f5\to 0}{\mathrm{log}}_{\u03f5}N(\u03f5)$. 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 |