Space-time Singularities vs. Topologies in the Zeeman—Göbel Class

We first observe that the Path topology of Hawking, King and MacCarthy is an analogue, in curved space-times, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski space-time. We then review a result of a recent paper on spaces of paths and the Path topology, and see that there are at least five more topologies in the class $$\mathfrak{Z}-\mathfrak{G}$$ of Zeeman-Gobel topologies which admit a countable basis, incorporate the causal and conformal structures, but the Limit Curve Theorem (LCT) fails to hold. The “problem” that the LCT does not hold can be resolved by “adding back” the light cones in the basic-open sets of these topologies, and create new basic open sets for new topologies. But, the main question is: do we really need the LCT. to hold, and why? Why is the manifold topology, under which the group of homeomorphisms of a space-time is vast and of no physical significance (Zeeman), more preferable than an appropriate topology in the class $$\mathfrak{Z}-\mathfrak{G}$$ under which a homeomorphism is an isometry (Gobel)? Since topological conditions that come as a result of a causality requirement are key in the existence of singularities in general relativity, the global topological conditions that one will supply the space-time manifold might play an important role in describing the transition from a quantum nonlocal theory to a classical local theory.