# Difference between revisions of "Metric space"

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A '''metric space''' is a pair, <math>(S, d)</math> of a [[set]] <math>S</math> and a [[metric]] <math>d: S \times S \to \mathbb{R}_{\geq 0}</math>. The metric <math>d</math> represents a distance function between pairs of points of <math>S</math>, and we require it to obey symmetry (<math>d(x, y) = d(y, x)</math>), positivity (<math>d(x, y) \geq 0</math> and <math>d(x, y) = 0</math> if and only if <math>x = y</math>) and the [[triangle inequality]] (<math>d(x, y) + d(y, z) \geq d(x, z)</math> for all points <math>x, y, z \in S</math>). | A '''metric space''' is a pair, <math>(S, d)</math> of a [[set]] <math>S</math> and a [[metric]] <math>d: S \times S \to \mathbb{R}_{\geq 0}</math>. The metric <math>d</math> represents a distance function between pairs of points of <math>S</math>, and we require it to obey symmetry (<math>d(x, y) = d(y, x)</math>), positivity (<math>d(x, y) \geq 0</math> and <math>d(x, y) = 0</math> if and only if <math>x = y</math>) and the [[triangle inequality]] (<math>d(x, y) + d(y, z) \geq d(x, z)</math> for all points <math>x, y, z \in S</math>). | ||

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+ | ==Popular metrics== | ||

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+ | * The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance | ||

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+ | * The [[discrete metric]] on any set | ||

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## Revision as of 16:56, 23 September 2006

A **metric space** is a pair, of a set and a metric . The metric represents a distance function between pairs of points of , and we require it to obey symmetry (), positivity ( and if and only if ) and the triangle inequality ( for all points ).

## Popular metrics

- The Euclidean metric on , with the "usual" meaning of distance

- The discrete metric on any set

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