A function, d, on pairs from a set S such that for each x, y, z in S: d(x, y) = 0; d(x, y) = 0 == x=y; d(x, y) + d(y, z) = d(x, z). The function is also called a distance; the pair (S, d) is a metric space. Condition (3) is called the Triangle inequality. A space is metrizable if a metric can be defined. A metric is induced by a norm (when it exists): d(x, y)=|| x - y||. See Hausdorff metric.