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.
They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.
Specifically, suppose that
are metric spaces and
Thus we have a metric map when, for any points
denote the metrics on
Consider the metric space
with the Euclidean metric.
is a metric map, since for
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.
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.
to the family of nonempty subsets of
is said to be Lipschitz if there exists
{\displaystyle H(Tx,Ty)\leq Ld(x,y),}
is the Hausdorff distance.
is called nonexpansive, and when
is called a contraction.