# Definition:Metric Space

## Contents

## Definition

A **metric space** $M = \left({A, d}\right)$ is an ordered pair consisting of:

together with:

- $(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms:

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

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

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

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

### Points of Metric Space

The elements of $A$ are called the **points** of the space.

### Distance Function

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

### Triangle Inequality

Axiom $M2$ is referred to as the **triangle inequality**, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.

## Notation

Some authors use the suboptimal $M = \left\{{A, d}\right\}$, which leaves it conceptually unclear as to which is the set and which the metric. This adds unnecessary complexity to the underlying axiomatic justification for the existence of the very object that is being defined.

The notation $M = \left[\!\left[{A, \rho}\right]\!\right]$ can also be found.

## Also see

- Pseudometric, which is the same as a metric but does not include the condition
**M4**.

- Quasimetric, which is the same as a metric but does not include the condition
**M3**.

- Results about
**metric spaces**can be found here.

### In Relation to Norms

- Metric Defines Norm iff it Preserves Linear Structure, where a homogeneous and translation invariant metric $d$ can be used to define a norm on a vector space.

- Metric Induced by Norm is Metric, where it is shown that any norm can be used to define a
**metric**:

- $d \left({x, y}\right) = \left\Vert {x - y} \right\Vert$

### In Relation to Topological Spaces

- For a
**metric space**$\left({A, d}\right)$, one can define the topology $\tau$ on $A$ induced by the metric $d$, thus making $\left({A, \tau}\right)$ a topological space. Thus topological notions carry over to**metric spaces**.

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.2$: Metric Spaces: Definition $2.1$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: The Definition - 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 5$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.1$: Motivation: Definition $2.1.2$ - 1999: Theodore W. Gamelin and Robert Everist Greene:
*Introduction to Topology*(2nd ed.) ... (previous) ... (next): $\S 1.1$: Open and Closed Sets