# Pointed version of Perelman stability theorem

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?

• See arxiv.org/abs/math/0002028 around Theorem 3.5. – Igor Belegradek Aug 17 at 22:49
• 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. – Igor Belegradek Aug 18 at 16:44