Let $X_i$ be n dimensional, no boundary Alexandrov spaces with curvature $\geqslant -1$ and diameter $\leqslant D$. Suppose that $X_i$ converge to an n dimensional Alexandrov space $X$. Then by Perelman's stability theorem, for i large, $X_i$ are homeomorphic to $X$.
Suppose $f:U\to R^m$ is regular on an open subset $U\subset X$, using approximation map $X\to X_i$, we can define a regular map $f_i:U_i\to R^m$. For $c\in R^m$, consider the level surfaces $f^{-1}(c), f_i^{-1}(c)$ with intrinsic metric, is $f_i^{-1}(c)$ homeomorphic to $f^{-1}(c)$ for i large?
A version of Stability theorem says that: there exists a homeomorphism $h_i: X\to X_i$ such that $f_i \circ h_i=f$ holds on every compact set $K\subset U$ and sufficiently large i. Then we have $h_i(f^{-1}(c))=f_i^{-1}(c)$. Since the intrinsic metrics of $f^{-1}(c), f_i^{-1}(c)$ are equivalent to the extrinsic metrics up to a constant, so $h_i$ is a homeomorphism from $f^{-1}(c)$ to $f_i^{-1}(c)$?
