Difference between revisions of "Metric"
(Importing text file) |
m |
||
Line 1: | Line 1: | ||
− | ''distance on a set | + | ''distance on a set $X$ '' |
− | A function | + | 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) | + | 1) $\rho(x,y)=0$ if and only if $x = y$ (the identity axiom); |
− | 2) | + | 2) $\rho(x,y) + \rho(y,z) \geq \rho(x,z)$ (the triangle axiom); |
− | 3) | + | 3) $\rho(x,y) = \rho(y,x)$ (the symmetry axiom). |
− | A set | + | 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 06:54, 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 |
Metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Metric&oldid=12195