Limit curve theorems for incomplete metric spaces and the null distance on Lorentzian manifolds

We prove a limit curve theorem for incomplete metric spaces. Our main application is to Sormani and Vegas’ null distance, where our results give strong control on the Lorentzian lengths of limit curves. We also show that regular cosmological time functions and the surface function of a Cauchy surface in a globally hyperbolic manifold define such a null distance.