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After a while I will accept one answer. The two answers are very interesting. BTW please read my meta post in this regards meta.mathoverflow.net/questions/1491/…
As a particular example I wonder which one (plain leaf space or holonomy groupoid) is more useful to distinguish the two foliations of the punctured plane defined by $e^{z^2}$ and $e^{z^3}$?
@LSpice I copy past the wikipedia text "Unfortunately, there is no universally accepted definition of a simple Lie group" for existence of several definitions. The same link contains definitions, references, etc
regarding the last paragraph of your post, as I wrote in my answer, the groupoid manifold is always Hausdorff (Of course in the reasonable and useful interpretation of leaf space, namely groupoid)
I would like to ask you to know if your question is answered by my answer or no?(at least in a partial manner). I would be very glad to see your feedback on this answer.
