  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 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-G\"obel
topologies which admit a countable basis, incorporate the causal and conformal
structures, but the Limit Curve Theorem fails to 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? Why is the manifold topology, under which the group of homeomorphisms
of a spacetime is vast and of no physical significance (Zeeman), more
preferable from an appropriate topology in the class
$\mathfrak{Z}-\mathfrak{G}$ under which a homeomorphism is an isometry
(G\"obel)? 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 spacetime manifold
might play an important role in describing the transition from the quantum
non-local theory to a classical local theory.
