metric space

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Related to metric space: Complete metric space

metric space

[′me·trik ′spās]
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.


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.


metric space

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 ?
Jachymski, The contraction principle for mappings on a metric space with a graph, Proc.
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Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, 2006.
5 Let (X,d) be a metric space, and let A [subset] X.
Non-Archimedean Alternative Contraction Principle) If (X, d) is a non-Archimedean generalized complete metric space and J: X [right arrow] X a strictly contractive mapping (that is d(J(x), J(y)) [greater than or equal to] Ld(y, x), for all x, y [member of] X and a Lipschitz constant L > 1), then either
A]) be two induced subobjects of the bounded metric space [A.
Pivot based algorithms consider the distances between an element and k preselected points, called "pivots", and maps the metric space on [R.
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Rakocevic: Some new extensions of Banach's contraction principle to partial metric space, Appl.
In the following we work with multimeasures defined on the Borel [sigma]-algebra F of a complete metric space [OMEGA].