I assume that for a Lorentzian manifold (i.e. with Minkowski signature), the analog of an open ball is the interior of a light cone. My question is motivated by the observation that whereas any point on the boundary of an open ball on a Rimannian manifold (i.e. with Euclidean signature) can be considered to be simultaneously interior to an infinite number of other open balls (and exterior to an infinite number of others) the boundary of a light cone is associated with a metric interval that is distinct from the timelike and spacelike intervals. For that reason, I wonder whether this introduces additional subtleties/restrictions on constructing a spacetime topology. Related to this question is under what circumstances (if any) can individual points be associated with certain kinds of intervals (e.g. Spacelike, timelike, null).
As Luboš says, there is no point in trying to define the topology (I assume that you are here trying to construct the topology, in its most rigrorous sense, from a base -- the set of open balls) via lightcones. The reason is that for a reasonable topology you should have arbitrary small(in some intuitive sense) open sets, so that the statements like "there exists an open set such that.." indeed mean what we want (i.e. "there exists a small enough open ball such that"). The lightcones are not in any sense small, and your topology will not see the continuity we are used to.
Well, one may say, forget about intuition, lets see where these open cones will lead us.
Now, I think that another question should be asked in this context: while a paracompact manifold always admits a positive-definite Riemannian metric, there is no such theorem for pseudo-Riemannian metrics. The reason is that you have to glue timelike curves from different coordinate charts in a sensible manner. What are the conditions for a manifold to admit a pseudo-Riemannian metric? I have found some usefull info here, but I do not have a complete answer.