Suppose [itex](P,M,\pi,G)[/itex] is a G-principal bundle. With this I mean a locally trivial fibration (G acts freely on P) over M=P/G with total space P and typical fibre G, as well as a differentiable surjective submersion [itex]\pi\colon P\to M[/itex]. In this case M is nearly a manifold, but may be non-Hausdorff. Now it is known that every principal bundle admits a connection if the base M is paracompact (this is the case if it is Hausdorff). My question is now if the converse does also hold. If I have a G-principal bundle with a G-invariant splitting of the tangent spaces of P into a vertical and horizontal part (or equivalently a connection one-form), does this imply that the base M must be Hausdorff? Any ideas how one could prove that? Or is it not true?