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

*d*_{fr}(*A*):=
− lim_{ε → 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*,

*d*_{H}(*A*):=
sup{*d* | *m*_{d}(*A*)
= ∞} = inf{*d* | *m*_{d}(*A*) = 0} ,

where *m*_{d} is
the *d*-dimensional outer measure of *A*.

* __Relationships__:
In general, *d*_{H}(*A*)
≤ *d*_{fr}(*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,

*d*_{info}(*x*):=
lim_{r → 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

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