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.