# Metric map

In the mathematical theory of metric spaces, a **metric map** is a function between metric spaces that does not increase any distance (such functions are always continuous).
These maps are the morphisms in the category of metric spaces, **Met** (Isbell 1964).
They are also called Lipschitz functions with Lipschitz constant 1, **nonexpansive maps**, **nonexpanding maps**, **weak contractions**, or **short maps**.

Specifically, suppose that *X* and *Y* are metric spaces and ƒ is a function from *X* to *Y*. Thus we have a metric map when, for any points *x* and *y* in *X*,

Here *d*_{X} and *d*_{Y} denote the metrics on *X* and *Y* respectively.

## Examples

## Category of metric maps

A map ƒ between metric spaces is an isometry if it is a bijective metric map whose inverse is also a metric map. The composite of metric maps is also metric. Thus metric spaces and metric maps form a category **Met**; **Met** is a subcategory of the category of metric spaces and Lipschitz functions, and the isomorphisms in **Met** are the isometries.

## Strictly metric maps

One can say that ƒ 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
from a metric space *X* to the family of nonempty subsets of *X* is said to be Lipschitz if there exists
such that

for all
, where *H* is the Hausdorff distance. When
, *T* is called nonexpansive and when
, *T* is called a contraction.

## See also

## References

- Isbell, J. R. (1964). "Six theorems about injective metric spaces".
*Comment. Math. Helv*.**39**: 65–76. doi:10.1007/BF02566944.