Short map

Definition

Suppose $(X,d_X)$ and $(Y,d_Y)$ are metric spaces. A function $f:X\to Y$ is termed a short map if it satisfies the following:

$\mathrm{\forall }a,b\in X,d_Y(f(a),f(b))\le d_X(a,b)$.

Note that any short map is a Lipschitz-continuous map and is hence also a uniformly continuous map.

Relation with other properties

Stronger properties

• Contraction

Weaker properties

• Lipschitz-continuous map
• Uniformly continuous map
• Continuous map