Natural vs. Artificial Topologies on a Relativistic Spacetime

Consider a set $M$ equipped with a structure $*$. We call a natural topology $T_*$, on $(M,*)$, the topology induced by $*$. For example, a natural topology for a metric space $(X,d)$ is a topology $T_d$ induced by the metric $d$ and for a linearly ordered set $(X,<)$ a natural topology should be the topology $T_<$ that is induced by the order $<$. This fundamental property, for a topology to be called "natural", has been largely ignored while studying topological properties of spacetime manifolds $(M,g)$ where $g$ is the Lorentz "metric", and the manifold topology $T_M$ has been used as a natural topology, ignoring the spacetime "metric" $g$. In this survey we review critically candidate topologies for a relativistic spacetime manifold, we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field, and give the foundations for future discussions. We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, G\"obel, Hawking-King-McCarthy and others, and we examine what should be meant by the term "natural topology" for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Theorem of Gao-Wald type of "time dilation" are such examples. }