Metric space

In mathematics, a metric space is, roughly speaking, a mathematical object which generalizes the notion of a Euclidean space $\mathbb {R} ^{n}$ which is equipped with the Euclidean distance to more general classes of sets, such as to a set of functions. A metric space consists of two components, a set and a metric on that set. On 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 on the set called the metric topology. If the associated set is also a vector space then the metric space becomes what is called a normed space.

Metric on 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:

$d(x_{1},x_{2})\geq 0\quad \forall x_{1},x_{2}\in X$ (non-negativity)
$d(x_{2},x_{1})=d(x_{1},x_{2})\quad \forall x_{1},x_{2}\in X$ (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)
$d(x_{1},x_{2})=0$ if and only if $x_{1}=x_{2}$

Formal 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 $O$ of subsets of $X$ (meaning that $A\in O\Rightarrow A\subset X$ ) 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 $O$ 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 simplest 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 $d_{E}(x,y)={\sqrt {\sum _{k=1}^{n}|x_{k}-y_{k}|^{2}}}\quad \forall x,y\in \mathbb {R} ^{n}$ .
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 $d(f,g)=\max _{x\in [a,b]}|f(x)-g(x)|\quad \forall f,g\in C[a,b]$ . Then $d$ is a metric on $C[a,b]$ and induces a topology on $C[a,b]$ often known as the norm topology or uniform topology.