{{Short description|Function between metric spaces that does not increase any distance}} In the [[mathematics|mathematical]] theory of [[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]]s in the [[category of metric spaces]], '''Met'''.{{r|isbell}} Such functions are always [[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>. In this example the Lipschitz constant is 1, that implies a metric map.
==Category of metric maps== The [[function composition]] of two metric maps is another metric map, and the [[identity map]] <math>\mathrm{id}_M\colon M \rightarrow M</math> on a metric space <math>M</math> is a metric map, which is also the [[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]] of the category of metric spaces and Lipschitz functions. A map between metric spaces is an [[isometry]] if and only if it is a [[bijective]] metric map whose [[Inverse function|inverse]] is also a metric map. Thus the [[isomorphism]]s in '''Met''' are precisely the isometries.
== Multivalued version == A mapping <math>T\colon 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}} * {{annotated link|Subcontraction map}}
==References==
<references>
<ref name=isbell>{{cite journal | author = Isbell, J. R. | authorlink = John R. Isbell | 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>
</references>
{{Metric spaces}} {{Topology}}
[[Category:Lipschitz maps]] [[Category:Metric geometry]] [[Category:Theory of continuous functions]]