# Difference between revisions of "Metric (analysis)"

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Together, the set <math>S</math> and the metric <math>d</math> form a [[metric space]]. | Together, the set <math>S</math> and the metric <math>d</math> form a [[metric space]]. | ||

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+ | Every metric space can be used to form a topology by considering taking the set of open balls as a [[topological basis]] (i.e. the sets <math>B_{x,\epsilon}:=\{y\in S:d(x,y)<\epsilon\}</math>). | ||

==Common metrics== | ==Common metrics== |

## Latest revision as of 19:15, 13 October 2019

A **metric** on a set is a function which obeys the following three properties:

- Symmetry: for all points .
- Positivity: for all and if and only if .
- The triangle inequality: for all .

Together, the set and the metric form a metric space.

Every metric space can be used to form a topology by considering taking the set of open balls as a topological basis (i.e. the sets ).

## Common metrics

- For , the Euclidean metric is the conventional distance function.

- For any set , the discrete metric and otherwise.

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