 Coarse embedding

A map $f:X\to Y$ between metric spaces is a coarse embedding when nondecreasing control functions ${\rho }_{-},{\rho }_{+}:[0,\infty )\to [0,\infty )$ satisfyThe last condition means that distances tending to infinity in the source must also tend to infinity in the target. The upper control prevents bounded source distances from having unbounded image distances. Unlike a metric embedding as defined here, a coarse embedding need not be injective: its lower control may vanish at small distances. Finite bi-Lipschitz distortion implies a coarse embedding by taking ${\rho }_{-}(t)=at$ and ${\rho }_{+}(t)=bt$, but nonlinear controls such as $\sqrt{t}$ allow much more compression.