In mathematical analysis, 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. 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.

is a function respectively.

Examples

with the Euclidean metric. 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 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

is the Hausdorff distance is called nonexpansive is called a contraction.

See also

References