# Definition:Metric Space/Distance Function

From ProofWiki

## Definition

Let $\left({A, d}\right)$ 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:

\((M1):\) | \(\displaystyle \forall x \in A:\) | \(\displaystyle d \left({x, x}\right) = 0 \) | |||||

\((M2):\) | \(\displaystyle \forall x, y, z \in A:\) | \(\displaystyle d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) \) | |||||

\((M3):\) | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle d \left({x, y}\right) = d \left({y, x}\right) \) | |||||

\((M4):\) | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle x \ne y \implies d \left({x, y}\right) > 0 \) |

## 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 $\left({A, d}\right)$ 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

- Bert Mendelson:
*Introduction to Topology*(1962)... (previous)... (next): $\S 2.2$: Metric Spaces - George McCarty:
*Topology: An Introduction with Application to Topological Groups*(1967)... (previous)... (next): $\text{III}$ - Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(1970)... (previous)... (next): $\text{I}: \ \S 5$ - W.A. Sutherland:
*Introduction to Metric and Topological Spaces*(1975)... (previous)... (next): $2.1$: Motivation: Definition $2.1.2$