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Let $(M,g)$ be a time-oriented smooth Lorentzian manifold, with Lorentzian metric $g$. In the following thread:

Lorentzian distance induced topology(a.k.a. Interval topology)

physicist @ValterMoretti makes the following claim:

"As a matter of fact, a (connected) Lorentzian smooth metric $g$ over the time-oriented smooth manifold $M$ does define a topology, the same already present on $M$."

I have not been able to find a proof of the statement above, unless some relatively strong assumptions are made on $(M,g)$ (for example imposing that $(M,g)$ is strongly causal). Can the statement be proved in general without further extra-assumptions? I was under the impression that in fact one of the key points of departure between Riemannian and Lorentzian geometry is the failure for a Lorentzian metric to define in general a topology equivalent to the pre-existing manifold topology, in contrast to what happens in Riemannian geometry.

Thanks.

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In general, this topology is coarser than the original topology of the manifold, and, without further assumptions, strictly coarser. It coincides with the original one iff the Lorentz manifold is strongly causal, see e.g. Prop 3.11 in Beem-Ehrlich-Easley. To get counter-example consider the cylinder $\mathbb{S}^1 \times \mathbb{R}$ with time direction being $\mathbb{S}^1$, i.e. periodic, and the usual flat metric. Then you can connect any two points by a timelike curve, thus the only non-empty open diamond is the whole spacetime. In this case the induced topology is the in-discrete one.

Addendum: Already in this example one see that the topology needs not to be Hausdorff, let alone metric.

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    $\begingroup$ A few remarks: (a) by "this topology" it is meant the Alexandrov/Interval topology defined using the base $I^+(x)\cap I^-(y)$. (b) This topology can be defined on any causal space and does not require strictly a smooth Lorentzian structure (nor, for that matter, a manifold structure on the base). (c) The previous fact should be taken as evidence toward why this topology does not have to coincide with the manifold topology. $\endgroup$
    – Willie Wong
    Commented Apr 11, 2017 at 14:13
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    $\begingroup$ @Willie Wong: yep, of course. This is the evidence from a more conceptual point of view. $\endgroup$
    – Stefan Waldmann
    Commented Apr 11, 2017 at 14:28
  • $\begingroup$ @StefanWaldmann Very nice answer. Is such topology in general(not only for the strongly causal spacetimes) metric? $\endgroup$
    – Bastam Tajik
    Commented Aug 16, 2023 at 7:37
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    $\begingroup$ @Bastam Tajik No, in general not. Already in the example I gave, it is not even Hausdorff. $\endgroup$
    – Stefan Waldmann
    Commented Nov 2, 2023 at 9:19
  • $\begingroup$ Ok let me ask a better question. Lorentzian distance is not an analytic metric. The question is what sort of Mathematical object is it precisely? @StefanWaldmann $\endgroup$
    – Bastam Tajik
    Commented May 21 at 15:31

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