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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} and all sets of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cup_{\gamma \in \Gamma}B_{r_{\gamma}}(x_{\gamma}),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} is an arbitrary index set (can be uncountable) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{\gamma}>0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\gamma} \in X} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \in \Gamma} . Then the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} satisfies all the requirements to be a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and is said to be the topology induced by the metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} endowed with the Euclidean distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_E} defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C[a,b]} of all real valued continuous functions on the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]\subset \mathbb{R}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<b} . Define the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d:C[a,b] \times C[a,b] \rightarrow \mathbb{R}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f,g)=\max_{x \in [a,b]}|f(x)-g(x)| \quad \forall f,g \in C[a,b]} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is a metric on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C[a,b]} and induces a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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