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 non-empty set. A metric on $\text{[math]}$ is a function

$\text{[math]}$

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

Non-negativity: $\text{[math]}$
Identity of indiscernibles: $\text{[math]}$
Symmetry: $\text{[math]}$
Triangle inequality: $\text{[math]}$ .

Properties

Property.

Non-negativity is implied by the other three properties.

Proof

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

$\text{[math]}$

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

$\text{[math]}$

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

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Reverse Triangle Inequality.

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