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.^[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 ${\displaystyle X}$, $${\displaystyle d_{Y}(f(x),f(y))\leq d_{X}(x,y).\!}$$ Here ${\displaystyle d_{X}}$ and ${\displaystyle d_{Y}}$ denote the metrics on ${\displaystyle X}$ and ${\displaystyle Y}$ respectively.