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].

**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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