# 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:

\((\text M 1)\) | $:$ | \(\displaystyle \forall x \in A:\) | \(\displaystyle \map d {x, x} = 0 \) | |||||

\((\text M 2)\) | $:$ | \(\displaystyle \forall x, y, z \in A:\) | \(\displaystyle \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | |||||

\((\text M 3)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle \map d {x, y} = \map d {y, x} \) | |||||

\((\text M 4)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle x \ne y \implies \map d {x, y} > 0 \) |

## 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

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.2$: Metric Spaces - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.1$: Motivation: Definition $2.1.2$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces - 1999: Theodore W. Gamelin and Robert Everist Greene:
*Introduction to Topology*(2nd ed.) ... (next):**One**: Metric Spaces: $1$: Open and Closed Sets - 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*... (previous) ... (next): $1.1$: Basic Definitions - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**metric (distance function)**