Dimension of a Space| Dimension of a Space |
In General
$ Def: A notion of dimension
is a map d: Top → \(\mathbb N\) ∪ {∞}, such that if
X ≅ Y then d(X) = d(Y),
and d(\(\mathbb R\)n)
= 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;
in Sakai 13.
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];
Georgiou et al T&A(12) [three types of invariants].
Small Inductive Dimension
$ Def: Defined inductively by
(1) ind(X) = −1 iff X = Ø;
(2) ind(X) ≤ n,
n ∈ \(\mathbb N\), 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(ε) is the smallest number of balls
of radius ε needed to cover A,
dfr(A):= − limε → 0 (ln N(ε) / ln ε) .
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 Adler 81, pp188 ff;
Urbański T&A(09) [transfinite Hausdorff dimension of a metric space];
Nicolini & Niedner PRD(11)-a1009 [of a quantum particle path with minimal length].
> Online resources:
see Wikipedia page.
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 → 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; Spectral Dimension.
* 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];
Smyth et al DM(10) [local topological dimension];
Calcagni et al PRD(15)-a1412 [dimensional flow];
> s.a. graph invariants [lattice, spectral
dimension, and others]; posets.
@ Generalizations:
Van Mill & Pol T&A(04) [splintered spaces];
Georgiou et al T&A(09) [dimension-like functions].
@ And physics: Brunner et al NJP(14)-a1401 [dimension of physical systems];
> s.a. models of dynamical spacetime; physical systems;
dimensionality of spacetime.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 3 oct 2019