  We prove that global hyperbolicity is stable in the interval topology on the
spacetime metrics. We also prove that every globally hyperbolic spacetime
admits a Cauchy hypersurface which remains Cauchy under small perturbations of
the spacetime metric. Moreover, we prove that if the spacetime admits a
complete timelike Killing field, then the light cones can be widened preserving
both global hyperbolicity and the Killing property of the field.
