Metric space

Lore & Background

The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples include a sphere equipped with angular distance and the hyperbolic plane. Metric spaces appear in many different branches of mathematics. For instance, Riemannian manifolds, normed vector spaces, and graphs may be viewed as metric spaces. In abstract algebra, the field of p-adic numbers is the completion of the rational numbers with respect to a certain metric. Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces. Many notions of analysis, including balls, completeness, and uniform, Lipschitz, and Hölder continuity, can be defined for metric spaces. Other notions such as continuity, compactness, and open and closed sets can be defined for metric spaces, but also in the even more general setting of topological spaces.

Reader's Guide

The concept of a metric space is fundamental in modern mathematics because it formalizes the intuitive idea of distance with minimal axioms: distance from a point to itself is zero, distinct points have positive distance, symmetry, and the triangle inequality. This generality gives metric spaces flexibility while encoding many intuitive facts about distance. As a result, general results about metric spaces can be applied in many different contexts, from geometry to analysis to algebra. A particular metric may not be best thought of as measuring physical distance, but instead as the cost of changing from one state to another (as with Wasserstein metrics) or the degree of difference between two objects (such as the Hamming distance between strings or the Gromov–Hausdorff distance between metric spaces themselves). The Euclidean plane can be equipped with many different metrics, including the familiar Euclidean distance, the taxicab or Manhattan distance, and the Chebyshev distance, each useful for different purposes.

Did You Know?

• The distance from a point to itself is always zero in a metric space.
• The triangle inequality states that the direct distance between two points is never longer than a path through a third point.
• The taxicab distance on the Euclidean plane is defined as the sum of the absolute differences of coordinates.