Dimension of a Space  

In General
$ Def: A notion of dimension is a map d: Top → \(\mathbb N\) ∪ {∞}, such that if XY 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 xX, G open neighborhood of x, ∃UG 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].

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 quantum spacetime.


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