Causality and topology in sub-Lorentzian geometry

Suppose that (M, H, g) is a sub-Lorentzian manifold, and let f: M → M be a bijection such that f and f−1 preserve timelike future directed curves. It is proved that under some mild assumptions made on (M, H, g), f is a homeomorphism. The presented result is a generalization of the known result in spacetime geometry obtained by Malament [J. Math. Phys. 18(7), 1399–1404 (1977)].