Let $(M^n,g\_i)$ be a sequence of smooth complete Riemannian manifold with $|sec\_{g\_i}| \le 1$. Suppose $(M\_i^n,g\_i)$ converges to a limit space $(X^{n-1},d)$ in the Gromov-Hausdorff sense, where the Hausdoff dimension of $X$ is $n-1$.


Can we show that $X$ contains no boundary point? Here, a point is a boundary point of $X$ if its tangent cone is isometric to $\mathbb R^{n-2} \times \mathbb R\_+$.



Does codimension-1 collapsing with bounded curvature have boundary?
Flat Klein bottles can collapse to a line segment.


