My first introduction to spacetime physics was Wheeler and Taylor's book Spacetime Physics. This book gave me an appreciation for how important the spacetime interval was for giving the distortions to space and time meaning. You know, the classical examples with the light beams and the trains, etc. So I have always associated the "spacetime interval" with the idea of distance.
However, I recently learned that, because a pseudo-Riemannian manifold can have zero length curves, it therefore is not a metric space by the specific definition mathematicians use. Now I am very confused. Is the concept of distance still useful to spacetime? Obviously it is in some sense, but now I'm not sure what sense I mean. Before I thought I might mean the metric space definition. But it would seem that metric spaces are not the right way to think about spacetime distance if a pseudo-Riemannian manifold isn't a metric space.
(1) What is a spacetime interval $ds^2$ mathematically? That is, what type of mathematical object is it?
(2) is the concept of a metric space useful for spacetime physics?