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Using the standard Whitney topologies on spaces of Lorentzian metrics, weshow that the existence of causal incomplete geodesics is a $C^\infty$-genericfeature within the class of spacetimes of a given dimension $n\geq 3$ that arestably causal, satisfy the timelike convergence condition (``strong energycondition'') and contain a codimension-two spacelike weakly trapped closedsubmanifold such as, e.g., a marginally outer trapped surface (MOTS). By usinga singularity theorem of Galloway and Senovilla for spacetimes containingtrapped closed submanifolds of codimension higher than two we also prove ananalogous $C^\infty$-genericity result for stably causal spacetimes with asuitably modified curvature condition and weakly trapped closed spacelikesubmanifold of any codimension $k> 2$.

Genericity of singularities in spacetimes with weakly trapped submanifolds