When we define a manifold we require that it locally looks like Euclidean. But even the Lorentzian metric in SR does not locally looks like Euclidean let alone the pseudo Riemannian metric used in GR. When, for example, the proper time vanishes as in a null path, the notion of an open ball, distance between events, and open basis cannot be used to define a topology.
Generally does a pseudo-Riemannian metric fit the definition of a manifold?