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 [Formula: see text] from a spacelike hypersurface [Formula: see text] 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.

中文翻译：

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闭锥结构中柯西超曲面和时间函数的规律性

我们在柯西超曲面上的一些结果补充了我们关于锥体上半连续分布的因果关系的工作。我们证明了每个局部稳定的非因果柯西超曲面都是稳定的。然后我们证明从一个类空间超曲面[公式：参见文本]的符号距离[公式：参见文本]在它的附近，与超曲面一样规则，并且通过使用这个事实，我们证明了每个柯西超曲面是与超曲面具有相同规律性的柯西时间（和陡峭）函数的水平集。我们还表明，在全局双曲闭锥结构中，具有边界的紧致类空间超曲面可以扩展到具有相同规律性的柯西类空间超曲面。我们以分离结果和密度结果结束工作。