# Does codimension-1 collapsing with bounded curvature have boundary?

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 $$\mathbb R^{n-2} \times \mathbb R_+$$.