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.
@ References:
Pasynkov T&A(08) [subset theorem].
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 > s.a. fractals.
$ For a (fractal) subset A of
Euclidean space: If N(
)
is the smallest number of balls of radius needed
to cover A,
dfr(A):= – limeps
to 0 (ln N()
/ ln ) .
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; Urbanski T&A(09) [transfinite Hausdorff
dimension of a metric space].
Information Dimension > s.a. spacetime
topology.
$ Def: If B(r,x)
is the ball of radius r centered at a point x in
phase space,
dinfo(x):= limr to 0 (ln V(B(r, x)) / ln r) ,
where 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.
* 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 references 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, and others]; posets.
@ Generalizations: Van Mill & Pol T&A(04)
[splintered spaces]; Georgiou et al T&A(09) [dimension-like functions].
@ For physical systems: Wolf & Perez-Garcia a0902 [quantum
systems, from evolution]; > s.a. models
of dynamical spacetime, quantum spacetime.
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