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:

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

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

1. 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 $x,y\in \mathbb {R} ^{n}$.
2. 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.
3. 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.

See also

Topology

Topological space

Normed space

References

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980