  In this paper we introduce and study the basic properties of de Rham
cohomology for a certain class of non-Hausdorff manifolds. After a careful
discussion of non-Hausdorff differential forms, we provide a description of de
Rham cohomology via Mayer-Vietoris sequences. We then use these sequences to
prove both de Rham's Theorem and the Gauss-Bonnet theorem for our non-Hausdorff
manifolds. Regarding the latter, we will see that the desired equality requires
additional counterterms coming from the geodesic curvature of certain
Hausdorff-violating submanifolds.
