Update. I have asked this on MO, but have not yet received an answer.
Proposition. The quotient map associated to a topological foliation (projecting to the leaf space) is open.
However the fibers of an open map from a topological manifold need not foliate its domain.
Example. Consider the multiplication map $\mathbb R^2\to \mathbb R,(x,y)\mapsto xy$. It is open, but I think its fibers do not give a topological foliation of the plane because the fiber over zero is the union of the axes, which does not look like parallel lines locally about the origin.
In the above example, the problem is at an isolated point. That leads me to wonder how far the fibers of an open map can be from "generically" foliations.
Definition. Let $X$ be a topological manifold. A partition of $X$ is generically a foliation of $X$ if it restricts to a foliation of an open dense subspace (which is also a topological manifold).
Example. I think the fibers of the multiplication maps give a generic foliation of the plane as they foliate the punctured plane.
I'm fairly certain a map (set function even) $f:X\to Y$ of topological spaces is open iff for every convergent net $y_\alpha\to y$ we have $f^{-1}(y)\subset\lim \left\{ f^{-1}(y_\alpha ) \right\}$, so the fibers of an open map are "stuck together" in this sense. I would like to understand exactly how different the latter sense is from the fibers actually being generically foliating.
Question. What are some examples of a (continuous) open map from a topological manifold $X$ whose fibers are not generically a foliation of $X$? What are some examples if we moreover assume the map is a quotient map?
Remark. I am looking for maximally "geometric" (i.e minimally "fractal") examples.
