Metric map

Consider the metric space [0,1/2] with the Euclidean metric. Then the function f(x)=x^2 is a metric map, since for x\ne y, |f(x)-f(y)|=|x+y||x-y|. 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 \mathrm{id}_M\colon M \rightarrow M on a metric space M 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 version ==

A mapping T\colon X\to \mathcal{N}(X) from a metric space X to the family of nonempty subsets of X is said to be Lipschitz if there exists L\geq 0 such that H(Tx,Ty)\leq L d(x,y), for all x,y\in X, where H is the Hausdorff distance. When L=1, T is called nonexpansive, and when L, T is called a contraction. ==See also==