# Difference between revisions of "Metric space"

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==Popular metrics== | ==Popular metrics== | ||

− | * The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance | + | * The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance which is given by <math>d(x,y)=\sum_{i=1}^n (x_i-y_i)^2</math> where <math>x=(x_1,x_2,\dots, x_n)</math> and <math>y=(y_1,y_2,\dots ,y_n)</math>. |

* The [[Discrete metric]] on any set, where <math>d(x,y)=1</math> if and only if <math>x\neq y</math> | * The [[Discrete metric]] on any set, where <math>d(x,y)=1</math> if and only if <math>x\neq y</math> |

## Revision as of 01:19, 22 December 2012

A **metric space** is a pair, of a set and a metric . The metric represents a distance function between pairs of points of which has the following properties:

- Symmetry: for all ,
- Non-negativity: for all ,
- Uniqueness: for all , if and only if
- The Triangle Inequality: for all points ,

Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.

## Popular metrics

- The Euclidean metric on , with the "usual" meaning of distance which is given by where and .

- The Discrete metric on any set, where if and only if

- The Taxicab metric on , with

*This article is a stub. Help us out by expanding it.*