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 $A$.


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

\((\text M 1)\)   $:$     \(\ds \forall x \in A:\) \(\ds \map d {x, x} = 0 \)      
\((\text M 2)\)   $:$   Triangle Inequality:      \(\ds \forall x, y, z \in A:\) \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)      
\((\text M 3)\)   $:$     \(\ds \forall x, y \in A:\) \(\ds \map d {x, y} = \map d {y, x} \)      
\((\text M 4)\)   $:$     \(\ds \forall x, y \in A:\) \(\ds x \ne y \implies \map d {x, y} > 0 \)      


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.


Some sources call a distance function just a distance, but that is a general term with a number of interpretations.


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 distance function, for example $\eth$.

Others use, for example, $\rho$.


Also see

  • Results about distance functions can be found here.


Sources

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