  We show that for a smooth manifold equipped with a singular Riemannian
foliation, if the foliated metric has positive sectional curvature, and there
exists a pre-section, that is a proper submanifold retaining all the transverse
geometric information of the foliation, then the leaf space has boundary. In
particular, we see that polar foliations of positively curved manifolds have
leaf spaces with nonempty boundary.
