Metric

The Metric System is a common term for the International System of Units.

In mathematics, a metric (also called distance metric) on a set is an abstraction of the notion of distance.

Intuitively, the distance between two point is a real, positive number, with the following properties:

• it is only zero if the two points are in fact the same
• it does not depend on the order in which points are given (the distance from London to Paris is the same as the distance from Paris to London)
• it satisfies the Triangle inequality: the distance between two points is never longer than going via a third intermediary point

Formally, let P be a set. A distance metric on P is a function mapping ordered pairs of elements of P, ( x , y ) to the positive Real numbers:

d : P × P -> R+

Furhermore, it must satisfy the following three axioms:

• d(p,q)=0 <=> p=q (the distance between two points is zero if and only if the two points are the same)
• d(p,q)=d(q,p) for all p,q in P (the distance between p and q is the same and the distance between q and p)
• d(p,q)+d(q,r) >= d(p,r) for all p,q,r in P (the sum of the distances between p and q and between q and r is greater than or equal to the distance between p and r)

Defining a metric on a set allows the notion of open sets limits to be defined on it: the metric d gives rise to a topology on P; a basis for this topology is the collection of sets { q: d(p,q)<e} for p in P and e>0.

In other words, a subset S of P is open in the metric topology if for every point p in S there is a number e>0 such that S contains the set { q: d(p,q)<e}.