  We complement our work on the causality of upper semi-continuous
distributions of cones with some results on Cauchy hypersurfaces. We prove that
every locally stably acausal Cauchy hypersurface is stable. Then we prove that
the signed distance $d_S$ from a spacelike hypersurface $S$ is, in a
neighborhood of it, as regular as the hypersurface, and by using this fact we
give a proof that every Cauchy hypersurface is the level set of a Cauchy
temporal (and steep) function of the same regularity as the hypersurface. We
also show that in a globally hyperbolic closed cone structure compact spacelike
hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces
of the same regularity. We end the work with a separation result and a density
result.
