Editing Metric space (section)

== External links ==
* {{Springer |title=Metric space |id=p/m063680}}
* [http://www.cut-the-knot.org/do_you_know/far_near.shtml Far and near—several examples of distance functions] at [[cut-the-knot]].

{{Metric spaces}}
{{Topology}}
{{Authority control}}

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A metric is called an [[ultrametric space|ultrametric]] if it satisfies the following stronger version of the ''triangle inequality'' for all <math>x,y,z\in X</math>:
:<math>d(x, y) \leq \max \{ d(x, z), d(y, z) \}.</math>

A metric ''<math>d</math>'' on a group ''<math>G</math>'' (written multiplicatively) is said to be {{em|left-invariant}} (resp. {{em|right-invariant}}) if for all <math>x, y, z \in G</math>
:<math>d(zx, zy) = d(x, y)</math> [resp. <math>d(xz,yz)=d(x,y)</math>].
A metric <math>d</math> on a commutative additive group <math>X</math> is said to be {{em|translation invariant}} if for all <math>x,y,z\in X</math>
:<math>d(x, y) = d(x + z, y + z),</math>or equivalently <math>d(x, y) = d(x - y, 0).</math>

== Examples ==
* The [[normed space]] <math>(\R, {|\cdot |})</math> is a [[Banach space]] where the absolute value is a [[Norm (mathematics)|norm]] on the real line <math>\R</math> that induces the usual [[Euclidean topology]] on <math>\R.</math> Define a metric <math>d : \R \times \R \to \R</math> on <math>\R</math> by <math>d(x, y) = {|\arctan(x) - \arctan(y)|}</math> for all <math>x,y\in\R.</math> Just like {{nowrap|<math>{|\cdot |}</math>{{hsp}}'s}} induced metric, the metric <math>d</math> also induces the usual Euclidean topology on <math>\R</math>. However, <math>d</math> is not a complete metric because the sequence <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> defined by <math>x_i := i</math> is a [[Cauchy sequence|{{nobr|<math>d</math>‑Cauchy}} sequence]] but it does not converge to any point of <math>\R</math>. As a consequence of not converging, this {{nobr|<math>d</math>-Cauchy}} sequence cannot be a Cauchy sequence in <math>(\R, {|\cdot |})</math> (i.e. it is not a Cauchy sequence with respect to the norm <math>{\|\cdot \|}</math>) because if it was {{nowrap|<math>| \cdot |</math>-Cauchy,}} then the fact that <math>(\R, {|\cdot |})</math> is a Banach space would imply that it converges (a contradiction).{{sfn|Narici|Beckenstein|2011|pp=47–51}}

== Equivalence of metrics ==

For a given set ''X'', two metrics <math>d_1</math> and <math>d_2</math> are called ''topologically equivalent'' (''uniformly equivalent'') if the identity mapping
:<math>{\rm id}: (X,d_1)\to (X,d_2)</math>
is a [[homeomorphism]] ([[uniform isomorphism]]).

For example, if <math>d</math> is a metric, then <math>\min (d, 1)</math> and <math>\frac{d}{1+d}</math> are metrics equivalent to <math>d.</math>

{{See also|Metric space#Notions of metric space equivalence}}

== References ==
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*{{citation
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William Lawvere
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*{{citation
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[[Category:Metric spaces| ]]
[[Category:Mathematical analysis]]
[[Category:Mathematical structures]]
[[Category:Topology]]
[[Category:Topological spaces]]
[[Category:Uniform spaces]]