Metric

distance on a set $X$

A function $\rho$ with non-negative real values, defined on the Cartesian product $X\times X$ and satisfying for any $x, y\in X$ the conditions:

1) $\rho(x,y)=0$ if and only if $x = y$ (the identity axiom);

2) $\rho(x,y) + \rho(y,z) \geq \rho(x,z)$ (the triangle axiom);

3) $\rho(x,y) = \rho(y,x)$ (the symmetry axiom).

A set $X$ on which it is possible to introduce a metric is called metrizable (cf. Metrizable space). A set $X$ provided with a metric is called a metric space.

Examples.

1) On any set there is the discrete metric \begin{equation} \rho(x,y) = 0 \text{ if } x=y \quad \text{and} \quad \rho(x,y) = 1 \text{ if } x\ne y. \end{equation}

2) In the space $\mathbb R^n$ various metrics are possible, among them are:

here .

3) In a Riemannian space a metric is defined by a metric tensor, or a quadratic differential form (in some sense, this is an analogue of the first metric of example 2)). For a generalization of metrics of this type see Finsler space.

4) In function spaces on a (countably) compact space there are also various metrics; for example, the uniform metric

(an analogue of the second metric of example 2)), and the integral metric

5) In normed spaces over a metric is defined by the norm :

6) In the space of closed subsets of a metric space there is the Hausdorff metric.

If, instead of 1), one requires only:

1') if (so that from it does not always follows that ), the function is called a pseudo-metric [2], [3], or finite écart [4].

A metric (and even a pseudo-metric) makes the definition of a number of additional structures on the set possible. First of all a topology (see Topological space), and in addition a uniformity (see Uniform space) or a proximity (see Proximity space) structure. The term metric is also used to denote more general notions which do not have all the properties 1)–3); such are, for example, an indefinite metric, a symmetry on a set, etc.

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2]	J.L. Kelley, "General topology" , Springer (1975)
[3]	K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
[4]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)

Comments

Potentially, any metric space has a second metric naturally associated: the intrinsic or internal metric. Potentially, because the definition may give for some pairs of points . One defines the length (which may be ) of a continuous path by , where is the infimum of all finite sums with a finite subset of which is an -net (cf. Metric space) and is listed in the natural order. Then is the infimum of the lengths of paths with , , but if there is no such path of finite length.

No reasonable topological restriction on suffices to guarantee that the intrinsic "metric" (or écart) will be finite-valued. If is finite-valued, suitable compactness conditions will assure that minimum-length paths, i.e. paths from to of length , exist. When every pair of points is joined by a path (non-unique, in general) of length , the metric is often called convex. (This is much weaker than the surface theorists' convex metric.) The main theorem in this area is that every locally connected metric continuum admits a convex metric [a1], [a2].

References

[a1]	R.H. Bing, "Partitioning a set" Bull. Amer. Math. Soc. , 55 (1949) pp. 1101–1110
[a2]	E.E. Moïse, "Grille decomposition and convexification" Bull. Amer. Math. Soc. , 55 (1949) pp. 1111–1121