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In the [[Mathematics|mathematical]] theory of [[Metric space|metric space]]s, a '''metric map''' is a [[Function (mathematics)|function]] between metric spaces that does not increase any distance. These maps are the [[Morphism|morphism]]s in the [[Category of metric spaces|category of metric spaces]], '''Met'''.{{r|isbell}} Such functions are always  [[Continuous function|continuous function]]s.
They are also called [[Lipschitz continuity|Lipschitz functions]] with Lipschitz constant 1, '''nonexpansive maps''', '''nonexpanding maps''', '''weak contractions''', or '''short maps'''.

Specifically, suppose that <math>X</math> and <math>Y</math> are metric spaces and <math>f</math> is a [[Function (mathematics)|function]] from <math>X</math> to <math>Y</math>. Thus we have a metric map when, for any points <math>x</math> and <math>y</math> in <math>X</math>,
<math display=block> d_{Y}(f(x),f(y)) \leq d_{X}(x,y) . \! </math>
Here <math>d_X</math> and <math>d_Y</math> denote the metrics on <math>X</math> and <math>Y</math> respectively.

== Examples ==

Consider the metric space <math>[0,1/2]</math> with the Euclidean metric. Then the function <math>f(x)=x^2</math> is a metric map, since for <math>x\ne y</math>, <math>|f(x)-f(y)|=|x+y||x-y|<|x-y|</math>.

==Category of metric maps==
The [[Function composition|function composition]] of two metric maps is another metric map, and the identity map <math>\mathrm{id}_M : M \rightarrow M</math> on a metric space <math>M</math> is a metric map, which is also the [[Identity element|identity element]] for function composition. Thus metric spaces together with metric maps form a [[Category (mathematics)|category]] '''[[Category of metric spaces|Met]]'''. '''Met''' is a [[Subcategory|subcategory]] of the category of metric spaces and Lipschitz functions. A map  between metric spaces is an [[Isometry|isometry]] if and only if it is a bijective metric map whose [[Inverse function|inverse]] is also a metric map. Thus the [[Isomorphism|isomorphism]]s in '''Met''' are precisely the isometries.

==Strictly metric maps==
One can say that <math>f</math> is '''strictly metric''' if the [[Inequality (mathematics)|inequality]] is strict for every two different points. Thus a [[Contraction mapping|contraction mapping]] is strictly metric, but not necessarily the other way around. Note that an isometry is ''never'' strictly metric, except in the [[Degeneracy (mathematics)|degenerate]] case of the [[Empty set|empty space]] or a single-point space.

== Multivalued version ==
A mapping <math>T:X\to \mathcal{N}(X)</math> from a metric space <math>X</math> to the family of nonempty subsets of <math>X</math> is said to be Lipschitz if there exists <math>L\geq 0</math> such that
<math display=block>H(Tx,Ty)\leq L d(x,y),</math>
for all <math>x,y\in X</math>, where <math>H</math> is the [[Hausdorff distance]]. When <math>L=1</math>, <math>T</math> is called nonexpansive and when <math>L<1</math>, <math>T</math> is called a [[Contraction mapping|contraction]].

==See also==

* {{annotated link|Contraction (operator theory)}}
* {{annotated link|Contraction mapping}}
* {{annotated link|Stretch factor}}

==References==

{{reflist|refs=

<ref name=isbell>{{cite journal
 | title = Six theorems about injective metric spaces
 | journal = Comment. Math. Helv.
 | volume = 39
 | year = 1964
 | pages = 65–76
 | url = http://www.digizeitschriften.de/resolveppn/GDZPPN002058340 | doi = 10.1007/BF02566944
}}</ref>

}}

{{Topology}}

[[Category:Theory of continuous functions]]
[[Category:Metric geometry]]

{{Sourceattribution|Metric map}}