Like any manifold, the pseudo-Riemannian manifold of spacetime in special or general relativity is a topological space, so there is a notion of open sets (or equivalently, neighborhoods) that allows us to talk about continuity, connectedness, etc. We implicitly use this structure whenever we frame the equivalence principle as saying that any spacetime "locally looks like Minkowski space" - the "locally" really means "in very small neighborhoods within the manifold". This point-set-topological structure is in a sense even more fundamental than anything relating to the metric, because any manifold has such a structure, whether or not is is pseudo-Riemannian (or even differentiable).
But what physically defines these open sets? For a Riemannian manifold (or more generally any metric space), in practice we always use the topology induced by the metric. But this doesn't work for a pseudo-Riemannian manifold, because the indefinite metric signature prevents it from being a metric space (in the mathematical sense). For example, if I emit a photon which "later" gets absorbed in the Andromeda Galaxy, then there is clearly a physical sense in which the endpoints of the null photon world line are "not infinitesimally close together", even though the spacetime interval separating them is zero (e.g. we could certainly imagine a physical field whose value varies significantly over the null trajectory). Is there a physical, coordinate- and Lorentz-invariant way to define the open sets of the spacetime?
(Note that I'm not talking about the global/algebraic topology of the spacetime, which is a completely separate issue.)