Difference between revisions of "Metric space"

(Formatting + brief comments on intuitive metric spaces.)
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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 (<math>d(x, y) = d(y, x)</math>)
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*Symmetry: for all <math>x, y \in S</math>, <math>d(x, y) = d(y, x)</math>
*Non-negativity (<math>d(x, y) \geq 0</math>
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*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 Metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance
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* The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance
  
* The [[Discrete Metric]] on any set
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* The [[Discrete metric]] on any set
  
 
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Revision as of 08:48, 30 November 2006

A metric space is a pair, $(S, d)$ of a set $S$ and a metric $d: S \times S \to \mathbb{R}_{\geq 0}$. The metric $d$ represents a distance function between pairs of points of $S$ which has the following properties:

  • Symmetry: for all $x, y \in S$, $d(x, y) = d(y, x)$
  • Non-negativity: for all $x, y \in S$, $d(x, y) \geq 0$
  • Uniqueness ($d(x, y) = 0$ if and only if $x = y$)
  • Triangle Inequality ($d(x, y) + d(y, z) \geq d(x, z)$ for all points $x, y, z \in S$).

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

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