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> which has the following properties: | 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> which has the following properties: | ||
| − | *Symmetry | + | *Symmetry: for all <math>x, y \in S</math>, <math>d(x, y) = d(y, x)</math> |
| − | *Non-negativity | + | *Non-negativity: for all <math>x, y \in S</math>, <math>d(x, y) \geq 0</math> |
*Uniqueness (<math>d(x, y) = 0</math> if and only if <math>x = y</math>) | *Uniqueness (<math>d(x, y) = 0</math> if and only if <math>x = y</math>) | ||
*[[Triangle Inequality]] (<math>d(x, y) + d(y, z) \geq d(x, z)</math> for all points <math>x, y, z \in S</math>). | *[[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== | ==Popular metrics== | ||
| − | * The [[Euclidean | + | * The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance |
| − | * The [[Discrete | + | * The [[Discrete metric]] on any set |
{{stub}} | {{stub}} | ||
Revision as of 08:48, 30 November 2006
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 (
if and only if 
) - Triangle Inequality (
for all points 
).
, 
if and only if 
)
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
, with the "usual" meaning of distance- The Discrete metric on any set
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