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Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.

In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold topology is strictly finer than the Alexandrov topology.

My question is that:

The necessary and sufficient conditions under which two points that are distinguishable under the Manifold topology, are indistinguishable under the Alexandrov topology for a generic such spacetime?

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I am not sure that is what you are really asking but if the Alexandrov topology is Hausdorff, it is the manifold topology and the spacetime is strongly causal. See for example Thm. 4.75 in

Minguzzi, E., Lorentzian causality theory, Living Rev. Relativ. 22, Paper No. 3, 202 p. (2019). ZBL1442.83021.

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  • $\begingroup$ Well, true that if the two topologies do not match then the Alexandrov topology is not Hausdorff. But "not being Hausdorff", doesn't mean that the two points are indistinguishable. There are weaker versions of distinguishability. I want the two points to share the same neighbourhood filter under Alexandrov topology and have non-equal neighbourhood filters under the manifold topology. Is this possible at all? $\endgroup$
    – Bastam Tajik
    Oct 20 at 15:47

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