There is an open subset $X_{0}\subset X$ such that $(X_{0},d_{X})$ is locally isometric to $(N_{0},\omega)$ where $N_{0}=N\backslash f(S)$, i.e. there is a homeomorphism $\phi:N_{0}\longrightarrow X_{0}$ such that, for any $y\in N_{0}$, there is a neighborhood $B_{y}\subset N_{0}$ of $y$ satisfying that, if $y_{1}$ and $y_{2}\in B_{y}$,

$$d_{\omega}(y_{1},y_{2})=d_{X}(\phi(y_{1}),\phi(y_{2})).$$

Furthermore, for any $y\in N_{0}$, there is a compact neighborhood $B\subset N_{0}$ and a holomorphic section $s:B\rightarrow f^{-1}(B)$, i.e., $f\circ s={\rm id}$, such that $s(y)\rightarrow\phi(y)$ under the Gromov-Hausdorff convergence of $(M,\tilde{\omega}_{t_{k}})$ to $(X,d_{X})$.