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.^[1] 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 ${\displaystyle X}$ and ${\displaystyle Y}$ are metric spaces and ${\displaystyle f}$ is a function from ${\displaystyle X}$ to ${\displaystyle Y}$. Thus we have a metric map when, for any points ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X}$,

$${\displaystyle d_{Y}(f(x),f(y))\leq d_{X}(x,y).\!}$$

${\displaystyle d_{X))$

${\displaystyle d_{Y))$

${\displaystyle X}$

${\displaystyle Y}$

Examples

Consider the metric space ${\displaystyle [0,1/2]}$ with the Euclidean metric. Then the function ${\displaystyle f(x)=x^{2))$ is a metric map, since for ${\displaystyle x\neq y}$, ${\displaystyle |f(x)-f(y)|=|x+y||x-y|<|x-y|}$.

Category of metric maps

The function composition of two metric maps is another metric map, and the identity map ${\displaystyle \mathrm {id} _{M}:M\rightarrow M}$ on a metric space ${\displaystyle M}$ 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.

Strictly metric maps

One can say that ${\displaystyle f}$ 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.

Multivalued version

A mapping ${\displaystyle T:X\to {\mathcal {N))(X)}$ from a metric space ${\displaystyle X}$ to the family of nonempty subsets of ${\displaystyle X}$ is said to be Lipschitz if there exists ${\displaystyle L\geq 0}$ such that

$${\displaystyle H(Tx,Ty)\leq Ld(x,y),}$$

${\displaystyle x,y\in X}$

${\displaystyle H}$

${\displaystyle L=1}$

${\displaystyle T}$

${\displaystyle L<1}$

${\displaystyle T}$