Distances and Metric Spaces

In General
* Remark on terminology: Following a common usage in physics, I will reserve the word "metric" for a metric tensor (or tensor field) g: V × V → \(\mathbb R\) on a vector space V, while the type of function d: X × X → \(\mathbb R\) on a set X defined below, that is often called "metric" in mathematics, will be called a "distance" here.
* Idea: A distance is the most common way of mathematically realizing the intuitive notion of closeness, although other definitions are possible.
$ Pseudometric space: A pair (X, d) with X a set and d a pseudodistance on X, a function d: X × X → \(\mathbb R\) satisfying (1) d(p, q) ≥ 0 and d(p, p) = 0, positive semi-definiteness; (2) d(p, q) = d(q, p), symmetry; (3) d(p, q) ≤ d(p, r) + d(r, q), triangle inequality; > s.a. types of topologies.
* Remark: In such a "degenerate metric space" two points x ≠ y with d(x, y) = 0 have to be equidistant from all other zs, because of the triangle inequality, and thus "indistinguishable"; This is not so in the Lorentzian case.
$ Distance: A positive-definite pseudodistance d: X × X → \(\mathbb R\).
$ Metric space: A pair (X, d), with d a distance on X.
* Relationships: It may arise from a norm; In the topology induced by the distance, a metric space is always paracompact.
@ References: Kurepa 63; Blumenthal 70; Schreider 74; Honig 95 [non-standard]; Deza & Deza 14 [encyclopedia].
@ And physics: Goodson 16 [dynamical systems].
> And other structures: see finsler geometry.

Related Notions and Results > s.a. cover; entropy; types of distances.
* Interesting maps: Isometries and continuous maps (in the induced topology) are not so rich; Distance-decreasing and λ-Lipschitz maps are more interesting.
* Baire category theorem: A complete metric space is not the countable union of nowhere-dense sets; This result can be stated as a theorem in Ramsey theory.
* Dilation of a map: For f : X → Y, dil f := sup_{x ≠ x'} d(f(x), f(x')) / d(x,x'); dil f := lim_{ε → 0} dil f |_B(x,ε).
* Diameter: If A ⊂ X is a subset of a metric space, its diameter is diam(A):= l.u.b.{d(x, y) | x, y in A}.
* Equicontinuity: A family \(\cal F\) of functions on a metric space (X, d) is equicontinuous iff for each ε > 0 there is a δ > 0 such that for all x and x' in X, and f in \(\cal F\), d(x, x') < δ implies |f(x)−f(x')| < ε.
$ Outer measure of a set: The d-dimensional outer measure of A ⊂ X is m_d(A):= lim_{ε → 0} inf ∑_i (diam S_i)^d, over all countable coverings of A by closed spheres S_i of diameter < ε.
@ Other structure on metric spaces: Parthasarathy 67 [probability measures]; Penot JGP(07) [tangent vectors and differentials of mappings].

New Distances out of Old
* Sum and sup: If d_i are distances on X (or even if all but one of them are pseudodistances), then two new distances on X are

d(x, y):= ∑_i a_i d_i(x, y), with a_i > 0 for all i ,   and   d(x, y):= sup_i d_i(x, y) .

* On subsets A of X: The induced distance d_A(a, b):= d_X(a, b) is always available; In addition, if d_X is induced by a length structure, we can choose to first induce a length structure on the subset, d_1,A(a, b):= inf_γ {l(γ) | a, b ∈ im(γ) ⊂ A}.
* On the Cartesian product of metric spaces: If (M₁, d₁) and (M₂, d₂) are metric spaces, then a metric on M₁ × M₂ is d((x₁, x₂),(y₁, y₂)):= sup_i d_i(x_i, y_i).

Space of Metric Spaces > s.a. distance between manifolds with metrics; Gromov-Hausdorff Space.
@ Measure: Kondo DG&A(05)

Generalizations
* Lorentzian metric space: A pair (M, d), with d: M × M → \(\mathbb R\)₊ ∪ {∞}, such that (i) d(x, y) > 0 implies d(y, x) = 0, so in particular d(x, x) = 0 for all x; (ii) d(x, y) d(y, z) > 0 implies that d(x, z) ≥ d(x, y) + d(y, z), the "reverse triangle inequality"; Examples of Lorentzian distance are the timelike geodesic distance between two points, or the volume of their Alexandrov set.
* Probabilistic metric space: A generalization of a metric space, where the distance has values in a set of probability distribution functions; > see Wikipedia page.
* Quantum metric space: A C*-algebra (or more generally an order-unit space) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric.
@ Lorentzian distance: Erkekoglu et al GRG(03) [level sets]; in Noldus CQG(04)gq/03; Rennie & Whale a1903 [finiteness and continuity]; > s.a. causality conditions; distance on a manifold with metric; world function.
@ Quantum metric space: Rieffel MAMS(04)m.OA/00 [Gromov-Hausdorff distance]; Latrémolière a1506 [Gromov-Hausdorff propinquity].
@ Other: Schweizer & Sklar 83 [probabilistic metric spaces]; Mizokami & Suwada T&A(05) [and their resolutions]; Kopperman et al T&A(09) [partial metric spaces, completion]; Antoniuk & Waszkiewicz T&A(11) [duality of generalized metric spaces].
> Bregman divergence: see Wikipedia page.

A qué le llaman distancia, eso me habrán de explicar − Atahualpa Yupanqui

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