A matchbox manifold is a generalized lamination, which is a continuum whose arccomponents define the leaves of a foliation of the space. The main result of this paper implies that a matchbox manifold which is manifold-like must be homeomorphic to a weak solenoid.
Clark, A., Hurder, S., & Lukina, O. (2017). MANIFOLD–LIKE MATCHBOX MANIFOLDS. Proceedings of the American Mathematical Society. doi:10.1090/proc/14200