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.



Metrically singular Alexandrov space.
**Suspicious example:**
Take a "funny" manifold with sectional curvature $\ge 1$ say $X$;
funny means Cayley flag or Aloff--Walach/Eschenburg/Bazaikin space, (not $S^n$ or $\mathbb{C}\mathrm{P}^n$ or $\mathbb{H}\mathrm{P}^n$).
Consider spherical suspension $\Sigma$ over $X$.


The space $\Sigma$ has curvature $\ge 1$.
It is expected that $\Sigma$ can not be approximated by any Riemannian manifolds with lower curvature bound, but no one can prove it.
The question has a lot in common with stabilized version of the Soul Theorem.


Partial results:


* Petersen--Wilhelm--Zhu: $\Sigma$ can not be approximated by Riemannian manifolds with curvature $\ge 1/4+\varepsilon$
* Kapovitch: for Cayley flag, if $\Sigma$ can be approximated then it is collapsing and the dimension drops down by 15 at least.


**P.S.**
It seems you forget to say "same dimension" in your questions.
In this case, in addition to Perelman's result you have a [result of V. Kapovitch](http://www.math.toronto.edu/vtk/noncolfin.pdf),
which says that space of directions, as well as iterated spaces of directions have to be homeomorphic to a sphere. This was conjectured by P. Petersen. 
(There are examples of Alexandrov spaces which homeomorphic to $\mathbb R^n$,
but space of directions at some point are not homeomorphic to spheres.)


