I'm curious about a exercise of Hirsch's book. Let $U$ be a open subset (nonempty) of $\mathbb{R}^2$. Suppose given a $C^r$ vector field on $U$ without zeros, such that each integral curve is closed in $U$. Let $M$ be the identification space obtained by collapsing each curve to a point. Then $M$ is a $C^r$ 1-manifold which can be non-Hausdorff.
I'm not intresed in solve this problem. I think that the Poincaré map helps to solve this, but I'm Curious how construct a non hausdorff space in this way. I'm trying to think about a vector field with close integral curves and no critical points on cylinder and vizualise this space But I can't. How this looks like?