  The class of Zeeman topologies on spacetimes in the frame of relativity
theory is considered to be of powerful intuitive justification, satisfying a
sequence of properties with physical meaning, such as the group of
homeomorphisms under such a topology is isomorphic to the Lorentz group and
dilatations, in Minkowski spacetime, and to the group of homothetic symmetries
in any curved spacetime. In this article we focus on two distinct topologies
that were suggested by Zeeman as alternatives to his Fine topology, showing
their connection with two orders: a timelike and a (non-causal) spacelike one.
For the (non-causal) spacelike order, we introduce a partition of the null cone
which gives the desired topology invariantly from the choice of the hyperplane
of partition. In particular, we observe that these two orders induce topologies
within the class of Zeeman topologies, while the two suggested topologies by
Zeeman himself are intersection topologies of these two order topologies
(respectively) with the manifold topology. We end up with a list of open
questions and a discussion, comparing the topologies with bounded against those
with unbounded open sets and their possible physical interpretation.
