In the same setting as Theorem 1.1, for any such limit space $(X,d_{X})$ there is an open dense subset $X_{0}\subset X$ such that $(X_{0},d_{X})$ is locally isometric to $(N\backslash f(S),\omega)$, i.e. there is a homeomorphism $\phi:N\backslash f(S)\longrightarrow X_{0}$ satisfying that, for any $y\in N\backslash f(S)$, there is a neighborhood $B_{y}\subset N\backslash f(S)$ of $y$ such that, for $y_{1}$ and $y_{2}\in B_{y}$,