TheInfoList

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.

An important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor allows distances along curves to be determined through integration, and thus determines a metric. However, not every metric comes from a metric tensor in this way.

## Definition

A metric on a set X is a function (called the distance function or simply distance)

$d:X\times X\to [0,\infty )$ ,

where $[0,\infty )$ is the set of non-negative real numbers and for all $x,y,z\in X$ , the following conditions are satisfied:

 1 $d(x,y)\geq 0$ non-negativity or separation axiom 2 $d(x,y)=0\Leftrightarrow x=y$ identity of indiscernibles 3 $d(x,y)=d(y,x)$ symmetry 4 $d(x,z)\leq d(x,y)+d(y,z)$ subadditivity or triangle inequality

Conditions 1 and 2 together define a positive-definite function. The first condition is implied by the others.

A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality where points can never fall 'between' other points:

$d(x,z)\leq \max(d(x,y),d(y,z))$ for all $x,y,z\in X$ A metric d on X is called intrinsic if any two points x and y in X can be joined by a curve with length arbitrarily close to d(x, y).

For sets on which an addition + : X × XX is defined, d is called a translation invariant metric if

$d(x,y)=d(x+a,y+a)$ for all x, y, and a in X.