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Perelman's stability theorem shows in particular that a finite dimensional compact Alexandrov space $(X,d)$ such that $X$ is not a topological manifold cannot be approximated in the Gromov-Hausdorf topology by Riemannian manifolds of the same dimension whose sectional curvature is bounded from below.

My questions are :

(1) Are there examples of compact Alexandrov spaces (say non-negatively curved) $(X,d)$ such that $X$ is a topological manifold but $(X,d)$ cannot be approximated by Riemannian manifolds of non-negative sectional curvature ?

(2) Does it change something if we allow the manifolds $(M_n,g_n)$ in the sequence to have a lower bound on the sectional curvature which is only going to zero as $n$ goes to infinity ?

Thanks.

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# Metrically singular Alexandrov space.

Perelman's stability theorem shows in particular that a finite dimensional compact Alexandrov space $(X,d)$ such that $X$ is not a topological manifold cannot be approximated in the Gromov-Hausdorf topology by Riemannian manifolds of the same dimension whose sectional curvature is bounded from below.

My questions are :

(1) Are there examples of compact Alexandrov spaces (say non-negatively curved) $(X,d)$ such that $X$ is a topological manifold but $(X,d)$ cannot be approximated by Riemannian manifolds of non-negative sectional curvature ?

(2) Does it change something if we allow the manifolds $(M_n,g_n)$ in the sequence to have a lower bound which is only going to zero as $n$ goes to infinity ?

Thanks.