My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a manifold like in this case.
But then it is hard not to wonder how a pseudo-Riemannian manifold, generalization of Riemannian manifolds, departs from the latter. I mean they are formally distinguished by the different metric tensor structure (indefinite vs definite positive) on top of the smooth manifold, but if the pseudo-Riemannian metric tensor is restricted by the base manifold topology to the space metric properties of a Riemannian metric I can only see left as difference between Riemannian and pseudo-Riemannian manifolds those referred locally to a manifold point's tangent space. But even there the manifold topology makes some restrictions AFAICS. Timelike or spacelike vectors at a point are allowed but no nonzero null vectors, and therefore no light-cone structure, if we are to go strictly by the $\Bbb R^4$ topology. I have read the wikipedia article on spacetime topology that refers to Zeeman and Hawking topologies, but those are not manifold topologies (they are not locally compact, nor metrizable), the other topologies mentioned there are equivalent to the manifold topology. Am I missing something here or is the above basically correct?
[Edit: The next two paragraphs centered on Minkowski space are easily answered just by considering it an affine space rather than a smooth manifold]
To be more specific when Zeeman writes in his paper of 1967 in Topology Vol. 6, 161-170 'The topology of Minkowski space': "LET M denote Minkowski space, the real 4-dimensional space-time continuum of special relativity. It is customary to think of M as having the topology of real 4-dimensional Euclidean space, although there are reasons why this is wrong. In particular:The 4-dimensional Euclidean topology is locally homogeneous, whereas M is not; every point has associated with it a light cone separating space vectors from time vectors."
Does this mean that the Euclidean topology, wich happens to be the same as the manifold topology, is incompatible with light cones, and if so how is Minkowski 4 dimensional manifold different from Euclidean 4-space, other than for the curvature invariant of the immersed hyperboloid preserved by the local isometry that gives rise to the indefinite form in 4 dimensions to begin with?