Metric space

In mathematics, a metric space is, roughly speaking, an abstract mathematical structure which 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 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:

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

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 $d_{E}(x,y)={\sqrt {\sum _{k=1}^{n}|x_{k}-y_{k}|^{2}}}\quad \forall x,y\in \mathbb {R} ^{n}$.
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 $d(f,g)=\max _{x\in [a,b]}|f(x)-g(x)|\quad \forall f,g\in C[a,b]$. 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.

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