# Difference between revisions of "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.

## Contents

### Examples.

1) On any set there is the discrete metric

2) In the space 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)