Template:Short description In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms in the category of metric spaces, Met.Template:R Such functions are always continuous functions. They are also called 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 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.
ExamplesEdit
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 mapsEdit
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 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 is also a metric map. Thus the isomorphisms in Met are precisely the isometries.
Multivalued versionEdit
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.