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 $\theta \le -c<0$ 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.