Difference between revisions of "Metric space"
m |
|||
| Line 1: | Line 1: | ||
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>). | ||
| + | |||
| + | ==Popular metrics== | ||
| + | |||
| + | * The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance | ||
| + | |||
| + | * The [[discrete metric]] on any set | ||
{{stub}} | {{stub}} | ||
Revision as of 17:56, 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
, with the "usual" meaning of distance- The discrete metric on any set
This article is a stub. Help us out by expanding it.
