  Recently, folk questions on the smoothability of Cauchy hypersurfaces and
time functions of a globally hyperbolic spacetime M, have been solved. Here we
give further results, applicable to several problems:
  (1) Any compact spacelike acausal submanifold H with boundary can be extended
to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples
to the smooth extension exist, but a continuous extension (in fact, valid for
any compact achronal subset K) is still possible.
  (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T
(i.e., a smooth function with past-directed timelike gradient everywhere, and
Cauchy hypersurfaces as levels) with S equal to one of the levels, is
constructed -thus, the spacetime splits orthogonally as $R \times S$ in a
canonical way. Even more, accurate versions of this result are obtained if the
Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but
non-acausal).
