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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?

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  • $\begingroup$ Not that it would be necessary for it to be a good question, but I think it would be interesting for the readers if you provided some info about why / in which context you feel the need of relaxing the Hausdorff hypothesis in smooth manifold theory. $\endgroup$
    – Qfwfq
    Commented Aug 24, 2011 at 21:54
  • $\begingroup$ I think the example with the most practical relevance is the leaf space of a foliation. Especially the 'arrows' space of a monodromy Lie-groupoid is a non Hausdorff manifold in many cases. $\endgroup$
    – Mirco
    Commented Aug 24, 2011 at 23:36
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    $\begingroup$ (1) All the standard induced bundles make perfectly good sense for non-Hausdorff manifolds. I don't know what you mean by your question (2), but it sounds too non-specific for this forum. (3) Have you tried a search? $\endgroup$
    – Ryan Budney
    Commented Aug 24, 2011 at 23:51
  • $\begingroup$ Re: 2) - I think it is too unfocussed a question, rather than too specialised. Regarding 1), it might be profitable to consider what happens in algebraic geometry, where Hausdorffness is not available: they deal with tangent bundles etc using maps from $k[x]/(x^2)$ (if you are working over a field $k$). Synthetic differential geometry might also be of use to you. In fact, I think SDG would the best way to approach this problem. $\endgroup$
    – David Roberts ♦
    Commented Aug 25, 2011 at 0:20
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    $\begingroup$ There's a standard construction of tangent bundles. The idea is that you can reconstruct your manifold as an adjunction of open subsets of Euclidean space (this works even for non-Hausdorff manifolds). So you can define the tangent bundle and any bundle construction provided you build the fibres in a functorial way from the tangent fibers. See for example Conlon's textbook. $\endgroup$
    – Ryan Budney
    Commented Aug 25, 2011 at 17:34

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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.

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  • $\begingroup$ Ok this a reference. Unfortunately it stays not in the realm of finite dimensional smooth manifolds. $\endgroup$
    – Mirco
    Commented Aug 25, 2011 at 12:12

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