A metric is any function d(x, y) that describes the distance between two points. Distance is formally defined as a single number with the following properties: (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x); (3) d(x, y) + d(y, z) > or = d(x, z) (the triangle inequality). The concept of a metric is important in differential geometry.
Metric space is a set that has a metric; in other words, a kind of space in which the concept of distance has meaning. Compare with topological space.