We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity $C^{1,1}$. The proof is based on regularisation techniques, combined with recent results in low regularity causality theory.
We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functi
ons on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional analytic foundations of these spaces.
