  We give a covariant definition of closeness between (time oriented)
Lorentzian metrics on a manifold M, using a family of functions which measure
the difference in volume form on one hand and the difference in causal
structure relative to a volume scale on the other hand. These functions will
distinguish two geometric properties of the Alexandrov sets $ A(p,q), \tilde{A}
(p,q) $ relative to two space time points $q$ and $p$ and metrics $g$ and $
\tilde{g} $. It will be shown that this family generates uniformities and
consequently a topology on the space of Lorentzian metrics which is Hausdorff
when restricted to strongly causal metrics. This family of functions will
depend on parameters for a volume scale, a length scale (relative to the volume
scale) and an index which labels a submanifold with compact closure of the
given manifold M.
