= Global causal step in the Penrose theorem
Assume future <null geodesic completeness>. The <compactness of the future horismos of a trapped surface> makes $E^+(S)$ a nonempty <compact> <achronal boundary> without edge. Projection along a complete <timelike vector field> into the <Cauchy hypersurface> is injective, open by <invariance of domain>, and closed by <compactness>. Connectedness forces surjectivity, contradicting the noncompact <Cauchy hypersurface>. This global step completes the contradiction.<Gravitational Collapse and Space-Time Singularities by Roger Penrose>[p. 59, compactness and noncompactness contradiction]{ref}
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