Dimension
of a Space| Dimension
of a Space |
In General
$ Def: A notion of dimension
is a map d: Top → N 
{
},
such that if X 
Y then d(X)
= d(Y), and d(Rn)
= n.
* Separable metrizable spaces:
Various possible dimension functions, e.g., covering dimension, small inductive
dimension, large inductive dimension; The main ones
coincide
and, for a linear space, give the number of elements of a basis.
* Non-metrizable spaces:
A satisfactory theory does not exist; even for compact spaces, only the Lebesgue
covering
dimension is really a theory.
* Other problems of study:
Sum theorems.
@ General references: in Eckmann & Ruelle RMP(85);
Manin BAMS(06)
[rev].
@ Texts: Hurewicz & Wallman 41 [classic; separable spaces]; Pears
75 [encyclopedic]; Engelking 78; Nagata 83 [general metric spaces]; in van
Mill
90.
Covering Dimension
$ For a topological space:
The least integer n such that every finite
open cover of X has an open refinement of order not exceeding n (infinite
if there is no such n), i.e., n + 1 is the minimum
number of elements of an
open cover that can be made to overlap.
* Relationships: For a space with both a linear and a topological
structure, the two definitions in general agree, but there are always pathological
cases.
Small Inductive Dimension
$ Def: Defined inductively
by
(1) ind(X) = –1 iff X = Ø;
(2) ind(X) 
n,
n 
N,
if for all x X,
G open neighborhood of x, 
U 
G
open, with ind(
U)
n–1;
(3) ind(X)
=
n if ind(X) n and
ind(X) > n–1.
* Special cases: ind(X)
= 0 iff X = Ø and it has an open
and closed topological basis.
Fractal or Capacity Dimension
$ For a fractal: The
fractal dimension is
dfr:= d ln M(R) / d ln R .
$ For a (fractal) subset A of Euclidean space:
dfr(A):= – limeps
to 0 (ln N(
)
/ ln ) ,
where N()
is the smallest number of balls of radius needed
to cover A.
Hausdorff Dimension > s.a. fractals [Mandelbrot
set]; random walk.
$ Def: For a set A contained in a metric space X,
dH(A):=
sup{d | md(A)
= } = inf{d | md(A)
= 0} ,
where md is the d-dimensional
outer measure of A.
* Relationships: In general, dH(A) dfr(A),
but they
often coincide [@ Barnsley].
@ References: in R Adler 81, pp188 ff.
Information Dimension > s.a. spacetime
topology.
$ Def: At a point x in
phase space,
dinfo(x):= limr to 0 (ln V(B(r, x)) / ln r) ,
where B(r,x) is the ball of radius r centered
at x, and
the measure V is
the fraction of time spent by the system in a region (if the system is ergodic,
this dimension is a.e. independent of x).
@ References: in Ruelle 89.
Other Definitions and Related Concepts > s.a. measure theory; models
of dynamical spacetime;
quantum spacetime.
* Global dimension of a ring R:
It is 0 if R is a field, 1 if it
is a principal
ideal domain.
@ Correlation dimension:
Ruelle PRS(90),
comment Essex & Nerenberg PRS(91),
and refs there.
@ Graph / discrete space: Evako IJTP(94)gq;
Nowotny & Requardt JPA(98)ht/97;
Reid
PRD(03)gq/02 [causal
set]; Bell & Dranishnikov T&A(08) [asymtotic dimension]; > s.a. graphs [lattice,
spectral
dimension]; posets.
@ Generalizations: Van Mill & Pol T&A(04) [splintered spaces].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
30 jun 2008