Genericity of singularities in spacetimes with weakly trapped submanifolds

Using the standard Whitney topologies on spaces of Lorentzian metrics, we show that the existence of causal incomplete geodesics is a $C^\infty$-generic feature within the class of spacetimes of a given dimension $n\geq 3$ that are stably causal, satisfy the timelike convergence condition (``strong energy condition'') and contain a codimension-two spacelike weakly trapped closed submanifold such as, e.g., a marginally outer trapped surface (MOTS). By using a singularity theorem of Galloway and Senovilla for spacetimes containing trapped closed submanifolds of codimension higher than two we also prove an analogous $C^\infty$-genericity result for stably causal spacetimes with a suitably modified curvature condition and weakly trapped closed spacelike submanifold of any codimension $k> 2$.