Theorem 8 Consider the set of smooth, periodic initial data to Equations (16)–(17) satisfying Equation (50). There is a subset of with the following properties

- is open with respect to the C
^{1}× C^{0}-topology on , - is dense with respect to the -topology on ,
- every spacetime corresponding to initial data in has the property that in one time direction, it is causally geodesically complete, and in the opposite time direction, the Kretschmann scalar is unbounded along every inextendible causal curve,
- for every spacetime corresponding to initial data in , the MGHD is C
^{2}-inextendible.

For the sake of completeness, let us also recall the definition of C^{2}-inextendibility.

Definition 9 Let be a connected Lorentz manifold, which is at least C^{2}. Assume there is a
connected C^{2} Lorentz manifold of the same dimension as and an isometric embedding
such that . Then is said to be C^{2}-extendible. If is not
C^{2}-extendible, it is said to be C^{2}-inextendible.

It might be possible to obtain this result using different methods. In fact, let be the set of initial data such that the corresponding solutions have an asymptotic velocity, which is different from one on a dense subset of the singularity. Endowing the initial data with the -topology, it is possible to show that is a dense set [77]. Due to its definition, it is clear that solutions corresponding to initial data in have the property that the curvature blows up on a dense subset of the singularity. It would be natural to expect the corresponding solutions to be inextendible, but providing a proof is nontrivial. Important steps in the direction of proving this statement were taken in [22]. However, to the best of our knowledge, there is, as yet, no result to this effect.

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