# Metric

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distance on a set A function with non-negative real values, defined on the Cartesian product and satisfying for any the conditions:

1) if and only if (the identity axiom);

2) (the triangle axiom);

3) (the symmetry axiom).

A set on which it is possible to introduce a metric is called metrizable (cf. Metrizable space). A set 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 , , or finite écart .

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.

How to Cite This Entry:
Metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Metric&oldid=12195
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article