  It is commonly known that in Riemannian and sub-Riemannian Geometry, the
metric tensor on a manifold defines a distance function. In Lorentzian
Geometry, instead of a distance function it provides causal relations and the
Lorentzian time-separation function. Both lead to the definition of the
Alexandrov topology, which is linked to the property of strong causality of a
space-time. We studied three possible ways to define the Alexandrov topology on
sub-Lorentzian manifolds, which usually give different topologies, but agree in
the Lorentzian case. We investigated their relationships to each other and the
manifold's original topology and their link to causality.
