  The structure of the moduli spaces $\M := \A/\G$ of (all, not just flat)
$SL(2,C)$ and $SU(1,1)$ connections on a n-manifold is analysed. For any
topology on the corresponding spaces $\A$ of all connections which satisfies
the weak requirement of compatibility with the affine structure of $\A$, the
moduli space $\M$ is shown to be non-Hausdorff. It is then shown that the
Wilson loop functionals --i.e., the traces of holonomies of connections around
closed loops-- are complete in the sense that they suffice to separate all
separable points of $\M$. The methods are general enough to allow the
underlying n-manifold to be topologically non-trivial and for connections to be
defined on non-trivial bundles. The results have implications for canonical
quantum general relativity in 4 and 3 dimensions.
