Metric Space

Synonyms

Distance space

Definition

In mathematics, a metric space is a pair M = (D, d), where D is a domain of objects (or objects’; keys or indexed descriptors) and d is a total (distance) function. The properties of the function d: D × D ↦ R, sometimes called the metric space postulates, are typically characterized as:

(p1)	$\forall x,y\in D,d\left(x,y\right)\ge 0$	Non-negativity,
(p2)	$\forall x,y\in D,d\left(x,y\right)=d\left(y,x\right)$	Symmetry,
(p3)	$\forall x\in D,d\left(x,x\right)=0$	Reflexivity,
(p4)	$\forall x,y\,{\in}\,D,x\,{\ne}\,y\,{\Rightarrow}\,d\left(x,y\right)\,{>}\,0$	Positiveness,
(p5)	$\begin{aligned}\forall x,y,z&{\in}\,D,d\left(x,z\right){\le}d\left(x,y\right)\\[-12pt]\end{aligned}$	Triangle inequality
	$\begin{aligned}\quad\quad\quad\quad+d\left(y,z\right)\end{aligned}$

Key Points

Modifying or even abandoning some of the metric function properties leads to interesting concepts that can better suit the reality in many situations. A pseudo-metric...