Graphics Reference
In-Depth Information
this set. It was popularized by B. Mandelbrot's work on fractals [Mand83]. So what
is a fractal? This leads us to the next topic.
22.3
Dimension Theory and Fractals
The concept of dimension, as obvious as it might seem, is something that only began
to be studied seriously in recent times. Of course, everyone is aware of the fact that
we seem to live in a three-dimensional world, but this idea remained vague, intuitive,
and was relatively unexplored until the late nineteenth century, although the concept
of n dimensions was introduced earlier in that century by A. Cayley, H. Grassmann,
and B. Riemann. At that time the concept meant quantities that needed, in some
unspecified way, a minimum of n real parameters to describe their points. Cantor's
discovery in 1878 that there was a bijective function between
R
and
R
2
showed that
“dimension” had nothing to do with the “number” of points involved. A few years later
in 1890, Peano proved the existence of a continuous map from [0,1] onto the unit
square [0,1] ¥ [0,1], so that “dimension” could in fact
not
be defined in terms of the
least number of continuous parameters required to describe a space. It was only at
the beginning of the twentieth century that satisfactory definitions of dimensions were
developed as a result of work of H. Poincaré, L.E.J. Brouwer, K. Menger, and P.
Urysohn. These definitions were essentially inductive definitions that involved looking
at the lower-dimensional subsets of a space that disconnect it. Many other definitions
of dimensions have been given since then. They are all equivalent for “nice” spaces
but can differ in other cases.
One of the first most basic results was the proof by Brouwer of the topological
invariance of what one calls the dimension of Euclidean space.
(Brouwer) If n π m , then
R
n
is not homeomorphic to
R
m
.
22.3.1
Theorem.
Proof.
This is Theorem 7.2.3.5(2) in [AgoM05].
Here is how dimension is defined in [HurW48]. One starts off defining the empty
space to have dimension -1 and then recursively defines the dimension of the space
to be the least integer n for which every point has arbitrarily small neighborhoods
whose boundaries have dimension less than n. More precisely,
Definition.
The
topological dimension
of a space
X
, denoted by dim
X
, is an integer
n satisfying the following conditions:
(1) The empty set and only it has dimension -1.
(2)
X
has dimension £ n ( n ≥ 0 ) at a point
p
if
p
has arbitrarily small neigh-
borhoods whose boundaries have dimension £ n - 1 .
(3)
X
has dimension £ n if
X
has dimension £ n at each of its points.
(4)
X
has dimension n at a point
p
if it is true that
X
has dimension £ n at
p
and
it is false that
X
has dimension £ n - 1 at
p
.
(5)
X
has dimension n if dim
X
£ n is true and dim
X
£ n - 1 is false.
(6)
X
has dimension • if dim
X
£ n is false for all n.
