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I am wondering if there is a version of the Perelman stability theorem which says approximately the following:

Let $\{(X_i,p_i)\}$ be a sequence of pointed $n$-dimensional complete Alexandrov spaces which converges in the Gromov-Hausdorff sense to an $n$-dimensional pointed complete Alexandrov space $(X,p)$. Then for any $R>0$ there exists $i(R)$ such that for all $i>i(R)$ the closed balls $B(p_i,R)\subset X_i$ are homeomorphic to the closed ball $B(p,R)\subset X$.

If this is not literally true, is there a modified version which is true?

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    $\begingroup$ See arxiv.org/abs/math/0002028 around Theorem 3.5. $\endgroup$
    – Igor Belegradek
    Commented Aug 17, 2019 at 22:49
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    $\begingroup$ Now the I have more time I wish to add that Perelman's proof is local in nature which is more or less why its "compact domain version" holds. Of course, there is no reason for $R$-balls to be homeomorphic. Rather the balls can be approximated by homeomorphic domains. $\endgroup$
    – Igor Belegradek
    Commented Aug 18, 2019 at 16:44

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