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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 anytopology on the corresponding spaces $\A$ of all connections which satisfiesthe weak requirement of compatibility with the affine structure of $\A$, themoduli space $\M$ is shown to be non-Hausdorff. It is then shown that theWilson loop functionals --i.e., the traces of holonomies of connections aroundclosed loops-- are complete in the sense that they suffice to separate allseparable points of $\M$. The methods are general enough to allow theunderlying n-manifold to be topologically non-trivial and for connections to bedefined on non-trivial bundles. The results have implications for canonicalquantum general relativity in 4 and 3 dimensions.Comment: Plain TeX, 7 pages, SU-GP-93/4-

Completeness of Wilson loop functionals on the moduli space of SL(2,C) and SU(1,1) connections