Difference between revisions of "Metric"
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''distance on a set
''distance on a set ''
A function with non-negative real values, defined on the Cartesian product and satisfying for any the conditions:
1) =0if 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|Metrizable space]]). A set provided with a metric is called a [[Metric space|metric space]].
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).
1) On any set there is the discrete metric
2) In the space various metrics are possible, among them are:
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:
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.
|||P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)|
|||J.L. Kelley, "General topology" , Springer (1975)|
|||K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)|
|||N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)|
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].
|[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|
Metric. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Metric&oldid=12195