Metric space

Introduction

In mathematics, a metric space is, roughly speaking, an abstract mathematical structure that generalizes the notion of a Euclidean space $\mathbb {R} ^{n}$ which has been equipped with the Euclidean distance, to more general classes of sets such as a set of functions. The notion of a metric space consists of two components, a set and a metric in that set. In a metric space, the metric replaces the Euclidean distance as a notion of "distance" between any pair of elements in its associated set (for example, as an abstract distance between two functions in a set of functions) and induces a topology in the set called the metric topology.

Metric in a set

Let $X\,$ be an arbitrary set. A metric $d\,$ on $X\,$ is a function $d:X\times X\rightarrow \mathbb {R}$ with the following properties:

$d(x_{2},x_{1})=d(x_{1},x_{2})\quad \forall x_{1},x_{2}\in X$ (symmetry)
$d(x_{1},x_{2})\leq d(x_{1},x_{3})+d(x_{3},x_{2})\quad \forall x_{1},x_{2},x_{3}\in X$ (triangular inequality)
$d(x_{1},x_{2})=0\ \Leftrightarrow \ x_{1}=x_{2}\,$

It follows from the above three axioms of a metric (also called distance function) that:

d(x_{1},x_{2})\geq 0\quad \forall x_{1},x_{2}\in X

(non-negativity)

Definition of metric space

A metric space is an ordered pair $(X,d)\,$ where $X\,$ is a set and $d\,$ is a metric on $X\,$ .

For shorthand, a metric space $(X,d)\,$ is usually written simply as $X\,$ once the metric $d\,$ has been defined or is understood.

Metric topology

A metric on a set $X\,$ induces a particular topology on $X\,$ called the metric topology. For any $x\in X$ , let the open ball $B_{r}(x)\,$ of radius $r>0\,$ around the point $x\,$ be defined as $B_{r}(x)=\{y\in X\mid d(y,x)<r\}$ . Define the collection ${\mathcal {O}}$ of subsets of $X\,$ (meaning that $A\in {\mathcal {O}}\Rightarrow A\subset X$ ) consisting of the empty set $\emptyset$ and all sets of the form:

\cup _{\gamma \in \Gamma }B_{r_{\gamma }}(x_{\gamma }),

where $\Gamma \,$ is an arbitrary index set (can be uncountable) and $r_{\gamma }>0\,$ and $x_{\gamma }\in X$ for all $\gamma \in \Gamma$ . Then the set ${\mathcal {O}}$ satisfies all the requirements to be a topology on $X\,$ and is said to be the topology induced by the metric $d\,$ . Any topology induced by a metric is said to be a metric topology.

Examples

The "canonical" example of a metric space, and indeed what motivated the general definition of such a space, is the Euclidean space $\mathbb {R} ^{n}$ endowed with the Euclidean distance $d_{E}\,$ defined by $d_{E}(x,y)={\sqrt {\sum _{k=1}^{n}|x_{k}-y_{k}|^{2}}}$ for all $x,y\in \mathbb {R} ^{n}$ .
Consider the set $C[a,b]\,$ of all real valued continuous functions on the interval $[a,b]\subset \mathbb {R}$ with $a<b\,$ . Define the function $d:C[a,b]\times C[a,b]\rightarrow \mathbb {R}$ by $d(f,g)=\max _{x\in [a,b]}|f(x)-g(x)|$ for all $f,g\in C[a,b]$ . This function $d\,$ is a metric on $C[a,b]\,$ and induces a topology on $C[a,b]\,$ often known as the norm topology or uniform topology.
Let $X\,$ be any nonempty set. The discrete metric on $X\,$ is defined as $d(x,y)=1\,$ if $x\neq y$ and $d(x,y)=0\,$ otherwise. In this case the induced topology is the so called discrete topology.