Metric space

Definition and axioms

Formally, a metric space is a pair (X,d) where X is a set and d is a distance function d:X×X→real numbers satisfying four conditions. In plain language these are:

• Non-negativity: distances are never negative.
• Identity of indiscernibles: the distance is zero exactly when the two points are the same.
• Symmetry: the distance from x to y equals the distance from y to x.
• Triangle inequality: the direct distance between two points is at most the sum of distances along a third point.

Typical examples

Common metrics that illustrate the concept include:

• The Euclidean distance on R^n, induced by the usual Pythagorean formula.
• The taxicab or l1 metric, summing coordinatewise absolute differences.
• The sup or l∞ metric, taking the largest coordinate difference.
• The discrete metric, which gives distance 1 for distinct points and 0 otherwise.
• Shortest-path distances on graphs, where the distance equals the minimal number of edges or minimal weighted length between vertices.

Structure and consequences

A metric determines a topology via open balls: for a point x and radius r>0 the ball consists of all points at distance less than r. This metric topology provides notions of continuity, convergence and compactness that mirror those in familiar geometric spaces. Important metric properties include completeness (every Cauchy sequence converges), separability (existence of a countable dense subset) and total boundedness, each with wide consequences in analysis.

Variations and related concepts

Several useful relaxations or strengthenings of the metric axioms appear in practice. A pseudometric allows distinct points to have zero distance; an ultrametric strengthens the triangle law to say the longest of two sides controls the third, producing unusual geometry with many equilateral triangles. Semimetrics and quasi-metrics drop or alter symmetry conditions. Embedding problems—whether one metric space can be mapped into another while approximately preserving distances—are central in geometry and theoretical computer science.

History, uses and significance

The abstract notion of metric spaces emerged in the early 20th century as mathematicians isolated the role of distance underlying analysis and topology. Since then metric concepts have become fundamental: they underpin convergence and continuity in real and functional analysis, provide the language for geometric group theory and fractal geometry, and appear in applied fields such as data analysis, clustering, and algorithms that depend on nearest-neighbor or shortest-path computations. Because metrics translate intuitive distance ideas into precise axioms, they serve as a bridge between concrete geometry and broad abstract theory.

Related facts and remarks

• Every metric space is a topological space, but not every topological space arises from a metric.
• Completeness depends on the metric chosen: the same set can be complete with one metric and incomplete with another.
• Metrics can be combined: sums, maxima, and certain integrals of metrics often yield new metrics on the same underlying set.