I found it very hard to find literature about smooth manifolds that are not required to be Hausdorff. In particular I'm interested in their local properties:


1.) Are the $r$-th order jet bundles $J^r(M,N)$ well defined for non Hausdorff manifolds?
(Recall that this question includes the tangent bundle as it is $J^1(\mathbb{R},M)$ at least for Hausdorff ones.)


2.) What are the basic consequences on the 'usual' structures on smooth manifolds, if we drop the Hausdorff assertion? (Like no partition of the unity ...)


3.) Is there a book or some other comprehensive work on non Hausdorff SMOOTH manifolds?



Jet spaces between non Hausdorff manifolds
It doesn't contain any proofs, but Bourbaki's *Variétés différentiables - Fascicule de résultats* defines jet bundles (Section 12) without assuming that the underlying varieties are Hausdorff.


