Metric map

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 , Met. 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 and are metric spaces and is a function from to . Thus we have a metric map when, for any points and in , Here and denote the metrics on and respectively. Consider the metric space with the Euclidean metric. Then the function is a metric map, since for , . The function composition of two metric maps is another metric map, and the identity map on a metric space is a metric map, which is also the identity element for function composition. Thus metric spaces together with metric maps form a . Met is a 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. One can say that is strictly metric if the inequality is strict for every two different points. Thus a contraction mapping is strictly metric, but not necessarily the other way around. Note that an isometry is never strictly metric, except in the degenerate case of the empty space or a single-point space. A mapping from a metric space to the family of nonempty subsets of is said to be Lipschitz if there exists such that for all , where is the Hausdorff distance. When , is called nonexpansive and when , is called a contraction.