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 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.

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 ${\displaystyle X}$,

Here ${\displaystyle d_{X}}$ and ${\displaystyle d_{Y}}$ denote the metrics on ${\displaystyle X}$ and ${\displaystyle Y}$ respectively.