metric space

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metric space

[′me·trik ′spās]
(mathematics)
Any topological space which has a metric defined on it.

Metric Space

 

a set of objects (points) in which a metric is introduced. Any metric space is a topological space; all possible open spheres are taken as neighborhoods in the space; in this case, the set of all points x for which the distance p(x, xo) < K is said to be an open sphere of radius K with center at the point XD- The topology of a given set may vary as a function of the metric introduced in it. For example, the following two metrics may be introduced in the set of real functions that are defined and continuous in the interval [a, b] of the number axis:

The corresponding metric spaces have different topological properties. A metric space with metric (1) is complete [for any sequence of its points {xn} such that ρ1(xn, xm) → 0 as n, m → ∞, we can find an element x0 of the metric space that is the limit of this sequence]; a metric space with metric (2) does not have this property. Fundamental concepts of analysis can be introduced in a metric space, such as the continuity of the mapping of one metric space into another, convergence, and compactness. The concept of metric space was introduced by M. Fréchet in 1906.

REFERENCES

Aleksandrov, P. S. Vvedenie v obshchuiu teoriiu mnozhestv i funktsii. Moscow-Leningrad, 1948.
Kolmogorov, A. N., and S. V. Fomin. Elementy teorii funktsii i funktsional’nogo analiza, 3rd ed. Moscow, 1972.
Liusternik, L. A., and V. I. Sobolev. Elementy funktsionaVnogo analiza, 2nd ed. Moscow, 1965.

V. I. SOBOLEV

metric space

(mathematics)
A set of points together with a function, d, called a metric function or distance function. The function assigns a positive real number to each pair of points, called the distance between them, such that:

1. For any point x, d(x,x)=0;

2. For any two distinct points x and y, d(x,y)>0;

3. For any two points x and y, not necessarily distinct,

d(x,y) = d(y,x).

4. For any three points x, y, and z, that are not necessarily distinct,

d(x,z) <= d(x,y) + d(y,z).

The distance from x to z does not exceed the sum of the distances from x to y and from y to z. The sum of the lengths of two sides of a triangle is equal to or exceeds the length of the third side.
References in periodicals archive ?
Haefliger, Metric spaces of non-positive curvature, Springer, 1999.
A) and (CLR) property in complex valued metric spaces, which extends and generalizes many results of the existing literature.
Bakhtin, "The contraction principle in quasi metric spaces," Journal of Functional Analysis, vol.
The first one presented in Section 3 treats a fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors.
Bojor, Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal.
The relation forms a partial order on the cone of metrics for a set: given any two metric spaces (X, [d.
Jachymski , IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J.
16] claimed that most of the coupled fixed point theorems in the setting of single valued mappings on ordered metric spaces are consequences of well-known fixed point theorems.
Let (X, d) be a metric space and consider a set valued map T on X with nonempty values in X.
In the realms of applied mathematics and materials science we find many recent applications of asymmetric metric spaces, for example, in rate-independent models for plasticity [1], shape-memory alloys [2], and models for material failure [3].
Let X be a geodesic metric space- that is, a metric space where any two points x and y are the endpoints of a curve of length d(x, y).
1985): Let (X,d) be a complete metric space and T: X [right arrow] X a contraction mapping with contraction factor c [member of] [0,1).