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I'm not in any doubt that Minkowski spacetime is globally hyperbolic, but can anyone provide or point to a proof of this, ideally starting with a proof it has the right differential structure before considerations of causal structure?

Removal of a single point is enough to destroy the causal structure, but does it also destroy the differential structure?

[As a side question: I was sure there was a question on SE somewhere which spoke of "$A \subset M$" being globally hyperbolic slab of Minkowski space but haven't been able to track it down - does anyone know of it?]

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  • $\begingroup$ The causal future and past of Minkowski space are just cones along the time axis, perhaps you could show that the intersection of any two such cones is a closed set? $\endgroup$ – Slereah Jan 16 '17 at 17:54
  • $\begingroup$ I wonder what it means to destroy a differential structure. $\endgroup$ – user130529 Jan 16 '17 at 18:09
  • $\begingroup$ @claudechuber Ceases to be differentiable, hence doesn't have a connection etc. $\endgroup$ – Julian Moore Jan 16 '17 at 18:17
  • $\begingroup$ @Slereah Oh, I might be able to manage that, I've done a few $I^{\pm}$ etc. pics in recent months ;) I was hoping/looking for something that someone else recognised as mathematically rigorous, i.e very formal so that I knew I hadn't missed some subtle point. And, IIRC, differential strcture comes before causality - is it OK to assume that since $M=\mathbb{R}^4$ the differentiability is given (and if I remove a point, it is lost)? $\endgroup$ – Julian Moore Jan 16 '17 at 18:22
  • $\begingroup$ @Julian Removing a point changes the topology, but I still don't see how it changes the differentiability, removing a point does not change the differential properties of the other points, differentiability is local. $\endgroup$ – user130529 Jan 16 '17 at 19:02

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