Limit Curve Theorem vs. Zeeman Topologies with a Countable Basis

In this article we first observe that the Path topology of Hawking, King and MacCarthy is an analogue in curved spacetimes, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski spacetime. We then review results of a recent paper on spaces of paths and the path topology, and see that there are at least five more, obvious, topologies in the class of Zeeman topologies, which admit a countable basis, incorporate the causal structure, but the Limit Curve Theorem does not hold. The "problem" that L.C.T. 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 L.C.T. to hold, and why?