Coarse embedding

A map $f:X\to Y$ is a coarse embedding when nondecreasing functions ${\rho }_{-},{\rho }_{+}:[0,\infty )\to [0,\infty )$ satisfy

A family $(X_n)$ uniformly coarsely embeds into $Y$ when maps $f_n:X_n\to Y$ admit the same two control functions ${\rho }_{-}$ and ${\rho }_{+}$ from the definition of a coarse embedding.