A seismologist and a pilot measure the distance between two places differently
Ask how far apart two points on Earth are and the answer depends on who is asking. Pilots and shipping lines want the shortest route over the surface, while seismologists care about the straight chord through the planet, which tracks how long quake waves take. Metric spaces are the mathematics that lets both answers count as distance.
A metric space is a set paired with a distance function, called a metric, that assigns a real number to every pair of points. The rules are few. A point is zero distance from itself, two different points are always a positive distance apart, distance is the same in both directions, and a detour through a third point can never beat the direct route, the triangle inequality. Those modest demands make the idea flexible yet strong enough to capture what distance intuitively means, so a theorem proved once applies in many settings.
The standard example is three-dimensional space with ordinary distance, and the real line with the absolute difference is the simplest; many ideas about metric spaces generalise real analysis and reduce to it on that line. A sphere measured by angle and the hyperbolic plane are others. The distance need not be physical at all. Among 100-character Unicode strings, the Hamming distance counts how many characters must change to turn one into another. Wasserstein metrics price the cost of shifting one distribution into another, and the Gromov-Hausdorff distance even measures how different two metric spaces are.
The concept runs through mathematics. Riemannian manifolds, normed vector spaces and graphs can all be viewed as metric spaces, and the p-adic numbers of abstract algebra arise by completing the rationals under a particular metric. Metric geometry and analysis on metric spaces study them for their own sake.
Much of analysis carries over directly. Balls, completeness and uniform, Lipschitz and Hölder continuity all need a metric to define. Ideas like compactness, continuity and open or closed sets work here too, though they also make sense in the broader setting of topological spaces, which drop distance altogether. By convention, people write just M for the pair of set and metric when the metric is clear.
Source: Metric space
