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 *z*s, 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 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

*

**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; > s.a.
causality conditions; 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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send feedback and suggestions to bombelli at olemiss.edu – modified 16
jun 2015