Reconstruction of mapping spaces by inverse limits

Extending the results of reconstruction of compact metric spaces by inverse limits, we show that if $(X,d),(Y,d)$ are compact metric spaces, then the mapping space $Y^X$ is homotopy equivalent to the inverse limit of an inverse system of finite $T_0$-spaces which depends only on the finite open covers of $X$ and $Y$. Applying our tools, we obtain that if $H$ is an isotopy of a compact metric space $(X,d)$, then $H_1{H}_0^{-1}$ can be approximated in terms of moves of a finite $T_0$-space.