# metric

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.