# 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.
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