# When does a leaf space admit a (non-Hausdorff) manifold structure?

If $f:M \to N$ is a submersion with connected fibers, then the fibers of $f$ foliate $M$. This is called a simple foliation of $M$ and the leaf space can be identified with $N$. Suppose that a foliation $\mathfrak{F}$ on $M$ has that the fundamental group of each leaf is finitely generated. Then one can show that $\mathfrak{F}$ is simple if and only the leaf space is Hausdorff and the holonomy group of each leaf is trivial.

Question 1: Is there a characterization of when a foliation $\mathfrak{F}$ is simple which works without demanding the above finitely generated condition on the fundamental groups of the leaves?

Note:

If $f:M \to N$ is any submersion (perhaps with disconnected fibers), then the connected componts of the fibers of $f$ foliate $M$. In this case, the leaf space can be identified with the etale-space of a sheaf over $N$, so has a (possibly non-Hausdorff) manifold structure.

Question 2:

Is there a characterization of when the leaf space of a foliation $\mathfrak{F}$ has the structure of a (possibly non-Hausdorff) manifold?

Remark: Simply removing the Hausdorff condition from the other characterization does not work, since e.g., the Kronecker foliation of the torus has no holonomy and each leaf is simply connected (diffeomorphic to $\mathbb{R}$ even), but the leaf space has the indiscrete topology.