Definition:Metric Space/Distance Function

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Definition

Let $\struct {A, d}$ be a metric space.


The mapping $d: A \times A \to \R$ is referred to as a distance function on $A$ or simply distance.


Here, $d: A \times A \to \R$ is a real-valued function satisfying the metric space axioms:

\((M1)\)   $:$     \(\displaystyle \forall x \in A:\) \(\displaystyle \map d {x, x} = 0 \)             
\((M2)\)   $:$     \(\displaystyle \forall x, y, z \in A:\) \(\displaystyle \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)             
\((M3)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle \map d {x, y} = \map d {y, x} \)             
\((M4)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle x \ne y \implies \map d {x, y} > 0 \)             


Also known as

The distance function $d$ is frequently referred to as a metric on $A$.

The two terms are used interchangeably on this website.


Also defined as

If $\struct {A, d}$ is a pseudometric space or quasimetric space, this definition still applies.

That is, a pseudometric and a quasimetric are also both found to be referred to in the literature as distance functions.


Also denoted as

Some authors use a variant of $d$ for a metric, for example $\eth$. Others use, for example, $\rho$.


Sources