dimension

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/Dimension2

dimension 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/

Cardinality

as 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

⁢

(

ϵ

)

be the minimal number of

open balls

in any covering of a bounded set

S

by balls of radius

ϵ

. The

http://planetmath.org/HausdorffDimension

Besicovitch-Hausdorff dimension of

S

is defined
as

-

lim

ϵ

→

0

⁡

log

ϵ

⁡

N

⁢

(

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

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