Let $(X,d_{X})$, $(Y,d_{Y})$ be two compact metric spaces. Suppose there exists maps $f:X\rightarrow Y$ and $g:Y\rightarrow X$ (not necessarily continuous) such that for all $x_{1},x_{2}\in X$,

$$|d_{X}(x_{1},x_{2})-d_{Y}(f(x_{1}),f(x_{2}))|<\epsilon$$

and for all $x\in X$,

$$d_{X}(x,g\circ f(x))<\epsilon,$$

and the two symmetric properties for $Y$ hold. Then we say the Gromov–Hausdorff distance between $X$ and $Y$ is at most $\epsilon$. The Gromov–Hausdorff distance $d_{GH}(X,Y)$ is the infinum of all such $\epsilon$.