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 ?
So let M be a separable negligible complete metric space which is a subset of an R-tree.
Bakhtin, "The contraction principle in quasi metric spaces," Journal of Functional Analysis, vol.
A direct approach is based on the framework of the metric space ([mathematical expression not reproducible]) (see [36]).
Jachymaski [3] introduced the notion of Banach G- contraction and proved some fixed point theorems for mappings satisfying this notion on complete metric space with a graph.
Any metric on n points can be embedded into the metric space [l.
Bojor, Fixed point of [psi]-contraction in metric spaces endowed with a graph, Ann.
Lakshmikantham and Ciric [11] proved coupled coincidence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results of Gnana Bhaskar and Lakshmikantham [8].
Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull.
Throughout this section (X, d) denotes an asymmetric metric space, unless the contrary is specified.
4 A CAT(0) space is a metric space having a unique geodesic between any two points, such that every triangle iy thin.
Theorem 4 ("Anti-collage theorem ", Vrscay and Saupe, 1999): 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).