  Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional
Riemannian Manifolds with uniform bounds on the sectional curvatures and let
(X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the
Gromov-Hausdorff sense. Let O \subseteq X be the set of points x \in X such
that there exists a neighborhood of x which is isometric to an open set in a
Riemannian orbifold and let B = O^c be the complement set. Then we have the
sharp estimates dim_Haus(B) \leq min{n-5, dim_Haus(X)-3}, and further for
arbitrary x \in X we have that x \in O iff a neighborhood of x has bounded
Alexandroff curvature. In particular, if n \leq 4 then B is empty and (X,d) is
a Riemannian orbifold. Our main application is to prove that a collapsed limit
of Einstein four manifolds has a smooth Riemannian orbifold structure away from
a finite number of points, and that near these points the curvatures has a
-dist^{-2} lower bound.
