Consider the open Möbius band $M$. This has a well-known foliation by circles, letting the central circle 'go around once' and all other circles go around twice.
Suppose we wish to consider the equivalence relation $$ \mathcal{R} = \{(x,y) \rvert \text{x and y lie on the same leaf} \} \subset M\times M.$$
It is known that this is not a smooth manifold. My problem is I can't really grasp what this object looks like. Apparently, it is the union of two $3$-manifolds, with a singular intersection (the two-manifold given by $L \times L$, with $L$ the central circle. (Which is why this can't be a smooth manifold.)
Does anyone have any tips on 'visualising' (or even just begin tackling) what this object looks like?
EDIT:I might have a clue where the two 3−manifolds come from. The foliation is due to the circle going around the manifold twice. I'm thinking that perhaps, we look at the restrictions x y if x and y lie on the same circle (going around once), and the restriction x y if x and y lie on the circle going the opposite direction. These will give two 3−manifolds (if they are even manifolds) whose singular intersection is L×L. –