TITLE: How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?
QUESTION [10 upvotes]: 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?

REPLY [4 votes]: There is a basic problem with the premise of the question as stated in the first paragraph. A pseudo-Riemannian(Lorentzian) metric tensor cannot produce a metric in the mathematical sense of a distance function(in semi-Riemannian manifolds the concept of length does not have a meaning), so it doesn't induce any topology, compatible or not, with the manifold topology.
The causal structure derived from the Lorentzian metric tensor can be used to define many different topologies according to the causality conditions of the specific Lorentzian manifold but unless they coincide with the manifold topology they can't replace the manifold topology(as explained by Andreas Cap above). See for example:"On the Topology of Lorentzian manifolds" by Renee Hoekzema.
The problems enters in relativity and physics in general when trying to relate measurable quantities or lengths, that are essential physically, with the causal structure topology in the context of a Lorentzian manifold.
Apparently the only topology coincident with a manifold topology is the Alexandrov topology in a strongly causal manifold. But this topology that is graphically depicted by two light cones united in the form of an infinitely elongated rhombus, paradoxically loses the causal light cone at that strongly causal limit, wich would render it equivalent to $E^4$ and erasing the distinction with a Riemannian manifold. 
This is actually realized in Minkowski space, that is strongly causal, and that is the only motive I see for authors trying to avoid treating Minkowski space as a manifold as it would stress this fact.
In the case of curved semi-Riemannian manifolds that are strongly causal the Hawking-Penrose singularity theorems apply.