It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. When we define a manifold we require that it locally looks like Euclidean. But even the Lorentzian metric in SR does not locally looks like Euclidean let alone the pseudo Riemannian metric used in GR. When, for example, the proper time vanishes as in a null path, the notion of an open ball, distance between events, and open basis cannot be used to define a toplogy. Does this imply that in cosmology, say through FLRW metric, we can only discuss the topology or global geometry of space, or spatial hypersurface, instead of spacetime? Also related to this question is that we know there "exists" a coordinate system in which the pseudo-Riemannian metric in GR becomes, locally, a Lorentzian one, thus having canonical signature - + + +. In FLRW metric we assume an isotropic and homogeneous cosmos based on observation in the, well, "observable" universe. But how about breaking this assumption and imagine that a global Riemannian metric or coordinates system "exists" for spacetime and only demand that it locally becomes Lorentzian? Which part of my understanding is correct and which one incorrect?