Let $\{(X_j,d_j)\}_{j=1}^{\infty}$ be a sequence of compact metric spaces, and let $(Y,d_Y)$ be a compact [metric space](/page/Metric%20Space). For a map $f_j:X_j\to Y$, define its distortion by
1. $d_{GH}(X_j,Y)\to 0$ as $j\to\infty$.
2. There exist $\varepsilon_j$-approximations $f_j:X_j\to Y$ with $\varepsilon_j\to 0$.
3. There exist correspondences $R_j\subset X_j\times Y$ with $\operatorname{dis}(R_j)\to 0$.