I disagree. A curve can be closed that is either timelike or spacelike. The interest in timelike curves is that it hypothetically can result in time travel wher

I disagree. A curve can be closed that is either timelike or spacelike. The interest in timelike curves is that it hypothetically can result in time travel whereas spacelike curves do not describe the trajectory of physical objects. Closed spacelike curves can be used to describe the boundary between causally connected events. Within the context of special relativity, an object travelling faster than light would be travelling along a spacelike curve, not a timelike curve. The significance of general relativity is that there are solutions that allow for the trajectory of an object to move along a closed timelike curve without exceeding the speed of light.