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''distance on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636201.png" />''
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''distance on a set $X$ ''
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636202.png" /> with non-negative real values, defined on the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636203.png" /> and satisfying for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636204.png" /> the conditions:
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636205.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636206.png" /> (the identity axiom);
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1) $\rho(x,y)=0$ if and only if $x = y$ (the identity axiom);
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636207.png" /> (the triangle axiom);
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2) $\rho(x,y) + \rho(y,z) \geq \rho(x,z)$ (the triangle axiom);
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636208.png" /> (the symmetry axiom).
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3) $\rho(x,y) = \rho(y,x)$ (the symmetry axiom).
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m0636209.png" /> on which it is possible to introduce a metric is called metrizable (cf. [[Metrizable space|Metrizable space]]). A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362010.png" /> provided with a metric is called a [[Metric space|metric space]].
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A set $X$ on which it is possible to introduce a metric is called metrizable (cf. [[Metrizable space|Metrizable space]]). A set $X$ provided with a metric is called a [[Metric space|metric space]].
  
 
===Examples.===
 
===Examples.===

Revision as of 08:54, 7 December 2012

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

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)


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
How to Cite This Entry:
Metric. Encyclopedia of Mathematics. URL: http://www.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