# metric space

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Related to metric space: Complete metric space

## 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).

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