Metric

In mathematics, a metric or distance function is a function that defines a rigorous concept of distance between each pair of elements in a given set. By generalizing the physical concept of distance, metrics allow mathematicians to define topological concepts such as open and closed sets, continuity, convergence, and limits in highly abstract spaces. A set equipped with a metric is called a metric space.

Definitions

Let $\text{[math]}$ be a given nonempty set. A metric on $\text{[math]}$ is a function

$\text{[math]}$

that satisfies the following four properties for all $\text{[math]}$ :

Nonnegativity: $\text{[math]}$

Identity of indiscernibles: $\text{[math]}$

Symmetry: $\text{[math]}$

Triangle inequality: $\text{[math]}$ .

The value $\text{[math]}$ is called the distance between $\text{[math]}$ and $\text{[math]}$ .

Properties

Property.

Nonnegativity is implied by the other three properties.

Proof

Let $\text{[math]}$ . By the triangle inequality,

$\text{[math]}$

By the identity of indiscernibles, $\text{[math]}$ . Thus

$\text{[math]}$

By symmetry, $\text{[math]}$ . Therefore,

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Hence $\text{[math]}$ .

□

Reverse triangle inequality.

Suppose that $\text{[math]}$ is a metric space and $\text{[math]}$ . Then

$\text{[math]}$

Proof

By the triangle inequality,

$\text{[math]}$

By symmetry, $\text{[math]}$ . Therefore,

$\text{[math]}$

Using the triangle inequality again,

$\text{[math]}$

Rearranging gives

$\text{[math]}$

Thus both

$\text{[math]}$

and

$\text{[math]}$

hold. Hence

$\text{[math]}$

□

Distance from a fixed point is continuous.

Let $\text{[math]}$ be a metric space and let $\text{[math]}$ . The function $\text{[math]}$ defined by

$\text{[math]}$

is continuous.

Proof

For any $\text{[math]}$ , the reverse triangle inequality gives

$\text{[math]}$

Let $\text{[math]}$ . If $\text{[math]}$ , then

$\text{[math]}$

Therefore $\text{[math]}$ is continuous.

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Open balls

Let $\text{[math]}$ be a metric space. For $\text{[math]}$ and $\text{[math]}$ , the open ball with centre $\text{[math]}$ and radius $\text{[math]}$ is the set

$\text{[math]}$

Similarly, the closed ball with centre $\text{[math]}$ and radius $\text{[math]}$ is the set

$\text{[math]}$

Open balls are fundamental because they describe the local structure of a metric space.

Metric topology

Every metric space naturally gives rise to a topology. A subset $\text{[math]}$ is called open if for every $\text{[math]}$ , there exists some $\text{[math]}$ such that

$\text{[math]}$

A subset $\text{[math]}$ is called closed if its complement $\text{[math]}$ is open.

This topology is called the metric topology induced by $\text{[math]}$ . Thus metrics allow concepts from analysis and topology to be studied in a common framework.

Convergence

Let $\text{[math]}$ be a metric space, and let $\text{[math]}$ be a sequence in $\text{[math]}$ . The sequence $\text{[math]}$ is said to converge to $\text{[math]}$ if

$\text{[math]}$

Equivalently, for every $\text{[math]}$ , there exists $\text{[math]}$ such that whenever $\text{[math]}$ ,

$\text{[math]}$

In this case, one writes

$\text{[math]}$

Uniqueness of limits.

In a metric space, a convergent sequence has at most one limit.

Proof

Suppose that $\text{[math]}$ and $\text{[math]}$ . By the triangle inequality,

$\text{[math]}$

Taking the limit as $\text{[math]}$ , both terms on the right tend to $\text{[math]}$ . Hence

$\text{[math]}$

By the identity of indiscernibles, $\text{[math]}$ . Therefore the limit is unique.

□

Continuity

Let $\text{[math]}$ and $\text{[math]}$ be metric spaces. A function $\text{[math]}$ is continuous at $\text{[math]}$ if for every $\text{[math]}$ , there exists $\text{[math]}$ such that whenever

$\text{[math]}$

one has

$\text{[math]}$

The function $\text{[math]}$ is called continuous if it is continuous at every point of $\text{[math]}$ .

This definition generalizes the usual epsilon delta definition of continuity from real analysis.

Examples

Standard metric on the real line

On $\text{[math]}$ , the usual metric is

$\text{[math]}$

This metric agrees with the ordinary notion of distance between real numbers.

Euclidean metric

On $\text{[math]}$ , the Euclidean metric is

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ .

Discrete metric

On any nonempty set $\text{[math]}$ , the discrete metric is defined by

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This metric treats every two distinct points as being exactly distance $\text{[math]}$ apart.

Taxicab metric

On $\text{[math]}$ , the taxicab metric is

$\text{[math]}$

In $\text{[math]}$ , this measures distance by moving only in coordinate directions, like travelling along the streets of a rectangular city grid.

Supremum metric

On $\text{[math]}$ , the supremum metric is

$\text{[math]}$

This metric measures the largest coordinate difference between two points.

Related notions

A pseudometric is similar to a metric, but it does not require that $\text{[math]}$ implies $\text{[math]}$ . Thus two distinct points may have distance $\text{[math]}$ .

A norm on a vector space gives rise to a metric by defining

$\text{[math]}$

For example, the usual absolute value on $\text{[math]}$ gives the standard metric

$\text{[math]}$

Definitions

Properties

Open balls

Metric topology

Convergence

Continuity

Examples

Standard metric on the real line

Euclidean metric

Discrete metric

Taxicab metric

Supremum metric

Related notions

See also