= Compactness of the future horismos of a trapped surface
In a future null-complete <globally hyperbolic spacetime> obeying the <null convergence condition>, the <future horismos> of a <compact> <trapped surface> is <compact>. Normalize its future null normals continuously using a timelike field. <Compactness> gives a uniform negative bound $\theta\leq-c<0$ on both initial <null expansions>. The <null focusing theorem> bounds all boundary generators by affine length $2/c$. Their endpoints lie in the image under a <continuous map> of the <compact> bundle of normalized null normals times $[0,2/c]$. The <future horismos> is <closed>, so is a <compact> subset of that image.
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